H
e
[Cajori1928].
Overview: Mathematical Markup Language (MathML) Version 2.0
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4 Content Markup
4.1 Introduction
4.1.1 The Intent of Content Markup
4.1.2 The Scope of Content Markup
4.1.3 Basic Concepts of Content Markup
4.2 Content Element Usage Guide
4.2.1 Overview of Syntax and Usage
4.2.2 Containers
4.2.3 Functions, Operators and Qualifiers
4.2.4 Relations
4.2.5 Conditions
4.2.6 Syntax and Semantics
4.2.7 Semantic Mappings
4.2.8 Constants and Symbols
4.2.9 MathML element types
4.3 Content Element Attributes
4.3.1 Content Element Attribute Values
4.3.2 Attributes Modifying Content Markup Semantics
4.3.3 Attributes Modifying Content Markup Rendering
4.4 The Content Markup Elements
4.4.1 Token Elements
4.4.2 Basic Content Elements
4.4.3 Arithmetic, Algebra and Logic
4.4.4 Relations
4.4.5 Calculus and Vector Calculus
4.4.6 Theory of Sets
4.4.7 Sequences and Series
4.4.8 Elementary classical functions
4.4.9 Statistics
4.4.10 Linear Algebra
4.4.11 Semantic Mapping Elements
4.4.12 Constant and Symbol Elements
As has been noted in the introductory section of this recommendation, mathematics can be distinguished by its use of a (relatively) formal language, mathematical notation. However, mathematics and its presentation should not be viewed as one and the same thing. Mathematical sums or products exist and are meaningful to many applications completely without regard to how they are rendered aurally or visually. The intent of the content markup in the Mathematical Markup Language is to provide an explicit encoding of the underlying mathematical structure of an expression, rather than any particular rendering for the expression.
There are many reasons for providing a specific encoding for content. Even a disciplined and systematic use of presentation tags cannot properly capture this semantic information. This is because without additional information it is impossible to decide if a particular presentation was chosen deliberately to encode the mathematical structure or simply to achieve a particular visual or aural effect. Furthermore, an author using the same encoding to deal with both the presentation and mathematical structure might find a particular presentation encoding unavailable simply because convention had reserved it for a different semantic meaning.
The difficulties stem from the fact that there are many to one mappings from presentation to semantics and vice versa. For example the mathematical construct ` H multiplied by e' is often encoded using an explicit operator as in H × e. In different presentational contexts, the multiplication operator might be invisible ` H e', or rendered as the spoken word `times'. Generally, many different presentations are possible depending on the context and style preferences of the author or reader. Thus, given ` H e' out of context it may be impossible to decide if this is the name of a chemical or a mathematical product of two variables H and e.
Mathematical presentation also changes with culture and time: some expressions in combinatorial mathematics today have one meaning to an Russian mathematician, and quite another to a French mathematician; see
Section 5.4.1 [Notational Style Sheets] for an example. Notations may lose currency, for example the use of musical sharp and flat symbols to denote maxima and minima
[Chaundy1954]. A notation in use in 1644 for the multiplication mentioned above was
H
e
[Cajori1928].
When we encode the underlying mathematical structure explicitly, without regard to how it is presented aurally or visually, we are able to interchange information more precisely with those systems that are able to manipulate the mathematics. In the trivial example above, such a system could substitute values for the variables H and e and evaluate the result. Further interesting application areas include interactive textbooks and other teaching aids.
The semantics of general mathematical notation is not a matter of consensus. It would be an enormous job to systematically codify most of mathematics - a task that can never be complete. Instead, MathML makes explicit a relatively small number of commonplace mathematical constructs, chosen carefully to be sufficient in a large number of applications. In addition, it provides a mechanism for associating semantics with new notational constructs. In this way, mathematical concepts that are not in the base collection of elements can still be encoded (Section 4.2.6 [Syntax and Semantics]).
The base set of content elements are chosen to be adequate for simple coding of most of the formulas used from kindergarten to the end of high school in the United States, and probably beyond through the first two years of college, that is up to A-Level or Baccalaureate level in Europe. Subject areas covered to some extent in MathML are:
It is not claimed, or even suggested, that the proposed set of elements is complete for these areas, but the provision for author extensibility greatly alleviates any problem omissions from this finite list might cause.
The design of the MathML content elements are driven by the following principles:
PCDATA or on additional processing such as operator precedence parsing.
The primary goal of the content encoding is to establish explicit connections between mathematical structures and their mathematical meanings. The content elements correspond directly to parts of the underlying mathematical expression tree. Each structure has an associated default semantics and there is a mechanism for associating new mathematical definitions with new constructs.
Significant advantages to the introduction of content-specific tags include:
Expressions described in terms of content elements must still be rendered. For common expressions, default visual presentations are usually clear. `Take care of the sense and the sounds will take care of themselves' wrote Lewis Carroll [Carroll1871]. Default presentations are included in the detailed description of each element occurring in Section 4.4 [The Content Markup Elements].
To accomplish these goals, the MathML content encoding is based on the concept of an expression tree. A content expression tree is constructed from a collection of more primitive objects, referred to herein as containers and operators. MathML possesses a rich set of predefined container and operator objects, as well as constructs for combining containers and operators in mathematically meaningful ways. The syntax and usage of these content elements and constructions is described in the next section.
Since the intent of MathML content markup is to encode mathematical expressions in such a way that the mathematical structure of the expression is clear, the syntax and usage of content markup must be consistent enough to facilitate automated semantic interpretation. There must be no doubt when, for example, an actual sum, product or function application is intended and if specific numbers are present, there must be enough information present to reconstruct the correct number for purposes of computation. Of course, it is still up to a MathML-compliant processor to decide what is to be done with such a content-based expression, and computation is only one of many options. A renderer or a structured editor might simply use the data and its own built-in knowledge of mathematical structure to render the object. Alternatively, it might manipulate the object to build a new mathematical object. A more computationally oriented system might attempt to carry out the indicated operation or function evaluation.
The purpose of this section is to describe the intended, consistent usage. The requirements involve more than just satisfying the syntactic structure specified by an XML DTD. Failure to conform to the usage as described below will result in a MathML error, even though the expression may be syntactically valid according to the DTD.
In addition to the usage information contained in this section, Section 4.4 [The Content Markup Elements] gives a complete listing of each content element, providing reference information about their attributes, syntax, examples and suggested default semantics and renderings. The rules for using presentation markup within content markup are explained in Section 5.2.3 [Presentation Markup Contained in Content Markup]. An informal EBNF grammar describing the syntax for the content markup is given in Appendix B [Content Markup Validation Grammar].
MathML content encoding is based on the concept of an expression tree. As a general rule, the terminal nodes in the tree represent basic mathematical objects, such as numbers, variables, arithmetic operations and so on. The internal nodes in the tree generally represent some kind of function application or other mathematical construction that builds up a compound object. Function application provides the most important example; an internal node might represent the application of a function to several arguments, which are themselves represented by the terminal nodes underneath the internal node.
The MathML content elements can be grouped into the following categories based on their usage:
These are the building blocks out of which MathML content expressions are constructed. Each category is discussed in a separate section below. In the remainder of this section, we will briefly introduce some of the most common elements of each type, and consider the general constructions for combining them in mathematically meaningful ways.
Content expression trees are built up from basic mathematical objects. At the lowest level,
leaf nodes are encapsulated in non-empty elements that define their type. Numbers and symbols are marked by the
token elements
cn and
ci. More elaborate constructs such as sets, vectors and matrices are also marked using elements to denote their types, but rather than containing data directly, these
container elements are constructed out of other elements. Elements are used in order to clearly identify the underlying objects. In this way, standard XML parsing can be used and attributes can be used to specify global properties of the objects.
The containers such as
<cn>12345<cn/> ,
<ci>x</ci> and
<csymbol definitionURL="mySymbol.htm" encoding="text">S</csymbol>represent mathematical numbers , identifiers and externally defined symbols. Below, we will look at
operator elements such as
plus or
sin, which provide access to the basic mathematical operations and functions applicable to those objects. Additional containers such as
set for sets, and
matrix for matrices are provided for representing a variety of common compound objects.
For example, the number 12345 is encoded as
<cn>12345</cn>
The attributes and
PCDATA content together provide the data necessary for an application to parse the number. For example, a default base of 10 is assumed, but to communicate that the underlying data was actually written in base 8, simply set the
base attribute to 8 as in
<cn base="8">12345</cn>
while the complex number 3 + 4i can be encoded as
<cn type="complex">3<sep/>4</cn>
Such information makes it possible for another application to easily parse this into the correct number.
As another example, the scalar symbol v is encoded as
<ci>v</ci>
By default,
ci elements represent elements from a commutative field (see
Appendix C [Content Element Definitions]). If a vector is intended then this fact can be encoded as
<ci type="vector">v</ci>
This invokes default semantics associated with the
vector element, namely an arbitrary element of a finite-dimensional vector space.
By using the
ci and
csymbol elements we have made clear that we are referring to a mathematical identifier or symbol but this does not say anything about how it should be rendered. By default a symbol is rendered as if the
ci or
csymbolelement were actually the presentation element
mi (see
Section 3.2.2 [Identifier (mi)]). The actual rendering of a mathematical symbol can be made as elaborate as necessary simply by using the more elaborate presentational constructs (as described in
Chapter 3 [Presentation Markup]) in the body of the
ci or
csymbol element.
The default rendering of a simple
cn-tagged object is the same as for the presentation element
mn with some provision for overriding the presentation of the
PCDATA by providing explicit
mntags. This is described in detail in
Section 4.4 [The Content Markup Elements].
The issues for compound objects such as sets, vectors and matrices are all similar to those outlined above for numbers and symbols. Each such object has global properties as a mathematical object that impact how they are to be parsed. This may affect everything from the interpretation of operations that are applied to them through to how to render the symbols representing them. These mathematical properties are captured by setting attribute values.
The notion of constructing a general expression tree is essentially that of applying an operator to sub-objects. For example, the sum
a +
b can be thought of as an application of the addition operator to two arguments
a and
b. In MathML, elements are used for operators for much the same reason that elements are used to contain objects. They are recognized at the level of XML parsing, and their attributes can be used to record or modify the intended semantics. For example, with the MathML
plus element, setting the
definitionURL and
encodingattributes as in
<plus definitionURL="www.vnbooks.com/VectorCalculus.htm"
encoding="text"/>
can communicate that the intended operation is vector-based.
There is also another reason for using elements to denote operators. There is a crucial semantic distinction between the function itself and the expression resulting from applying that function to zero or more arguments which must be captured. This is addressed by making the functions self-contained objects with their own properties and providing an explicit
apply construct corresponding to function application. We will consider the
apply construct in the next section.
MathML contains many pre-defined operator elements, covering a range of mathematical subjects. However, an important class of expressions involve unknown or user-defined functions and symbols. For these situations, MathML provides a general
csymbol element, which is discussed below.
apply constructThe most fundamental way of building up a mathematical expression in MathML content markup is the
apply construct. An
apply element typically applies an operator to its arguments. It corresponds to a complete mathematical expression. Roughly speaking, this means a piece of mathematics that could be surrounded by parentheses or
`logical brackets' without changing its meaning.
For example, (x + y) might be encoded as
<apply> <plus/> <ci> x </ci> <ci> y </ci> </apply>
The opening and closing tags of
apply specify exactly the scope of any operator or function. The most typical way of using
apply is simple and recursive. Symbolically, the content model can the described as:
<apply> op a b </apply>
where the
operands a and b are containers or other content-based elements themselves, and
op is an operator or function. Note that since
apply is a container, this allows
apply constructs to be nested to arbitrary depth.
An
apply may in principle have any number of operands:
<apply> op a b [c...] <apply>
For example, (x + y + z) can be encoded as
<apply> <plus/> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply>
Mathematical expressions involving a mixture of operations result in nested occurrences of
apply. For example,
a
x +
b would be encoded as
<apply>
<plus/>
<apply>
<times/>
<ci> a </ci>
<ci> x </ci>
</apply>
<ci> b </ci>
</apply>
There is no need to introduce parentheses or to resort to operator precedence in order to parse the expression correctly. The
apply tags provide the proper grouping for the re-use of the expressions within other constructs. Any expression enclosed by an
apply element is viewed as a single coherent object.
An expression such as (F + G)(x) might be a product, as in
<apply>
<times/>
<apply>
<plus/>
<ci> F </ci>
<ci> G </ci>
</apply>
<ci> x </ci>
</apply>
or it might indicate the application of the function F + G to the argument x. This is indicated by constructing the sum
<apply> <plus/> <ci> F </ci> <ci> G </ci> </apply>
and applying it to the argument x as in
<apply>
<apply>
<plus/>
<ci> F </ci>
<ci> G </ci>
</apply>
<ci> x </ci>
</apply>
Both the function and the arguments may be simple identifiers or more complicated expressions.
In MathML 1.0 , another construction closely related to the use of the
apply element with operators and arguments was the
reln element. The
relnelement was used to denote that a mathematical relation holds between its arguments, as opposed to applying an operator. Thus, the MathML markup for the expression
x <
y was given in MathML 1.0 by:
<reln> <lt/> <ci> x </ci> <ci> y </ci> </reln>
In MathML 2.0, the
apply construct is used with all operators, including logical operators. The expression above becomes
<apply> <lt/> <ci> x </ci> <ci> y </ci> </apply>
in MathML 2.0. The use of
reln with relational operators is supported
for reasons of backwards compatibility, but deprecated. Authors creating new content are
encouraged to use
apply in all cases.
The most common operations and functions such as
plus and
sin have been predefined explicitly as empty elements (see
Section 4.4 [The Content Markup Elements]). They have
type and
definitionURLattributes, and by changing these attributes, the author can record that a different sort of algebraic operation is intended. This allows essentially the same notation to be re-used for a discussion taking place in a different algebraic domain.
Due to the nature of mathematics the notation must be extensible. The key to extensibility is the ability of the user to define new functions and other symbols to expand the terrain of mathematical discourse.
It is always possible to create arbitrary expressions, and then to use them as symbols in the language. Their properties can then be inferred directly from that usage as was done in the previous section. However, such an approach would preclude being able to encode the fact that the construct was a known symbol, or to record its mathematical properties except by actually using it. The
csymbol element is used as a container to construct a new symbol in much the same way that
ci is used to construct an identifier. (Note that
`symbol' is used here in the abstract sense and has no connection with any presentation of the construct on screen or paper). The difference in usage is that
csymbol should refer to some mathematically defined concept with an external definition referenced via the
definitionURL attribute, whereas
ci is used for identifiers that are essentially
`local' to the MathML expression and do not use any external definition mechanism. The target of the
definitionURLattribute on the
csymbol element may encode the definition in any format: the particular encoding in use is given by the
encoding attribute
To use
csymbol to describe a completely new function, we write for example
<csymbol definitionURL="www.vnbooks.com/VectorCalculus.htm"
encoding="text">
Christoffel
</csymbol>
The
definitionURL attribute specifies a URI that provides a written definition for the
Christoffel symbol. Suggested default definitions for the content elements of MathML appear in
Appendix C [Content Element Definitions] in a format based on OpenMath, although there is no requirement that a particular format be used. The role of the
definitionURL attribute is very similar to the role of definitions included at the beginning of many mathematical papers, and which often just refer to a definition used by a particular book.
MathML 1.0 supported the use of the
fn to encode the fact that a construct is explicitly being used as a function or operator. To record the fact that
F+
G is being used semantically as if it were a function, it was encoded as:
<fn>
<apply>
<plus/>
<ci>F</ci>
<ci>G</ci>
</apply>
</fn>
This usage, although allowed in MathML 2.0 for reasons of backwards compatibility,
is now deprecated.
The fact that a construct is being used as an operator is clear from the position of the construct as the
first child of the
apply. If it is required to add additional information to the construct, it should be wrapped in a
semanticselement, for example:
<semantics definitionURL="www.mathslib.com/vectorfuncs/plus.htm"
encoding="Mathematica">
<apply>
<plus/>
<ci>F</ci>
<ci>G</ci>
</apply>
</semantics>
MathML 1.0 supported the use of
definitionURL with
fn to refer to external definitions for user-defined
functions. This usage, although allowed for reasons of backwards
compatibility, is deprecated in
MathML 2.0 in favour of using
csymbol to define the function, and then
apply to link the function to its arguments. For example:
<apply>
<csymbol definitionURL="http://www.defs.org/function_spaces.html#my_def"
encoding="text">
BigK
</csymbol>
<ci>x</ci>
<ci>y</ci>
</apply>
Given functions, it is natural to have functional inverses. This is handled by the
inverse element.
Functional inverses can be problematic from a mathematical point of view in that it implicitly involves the definition of an inverse for an arbitrary function F. Even at the K-through-12 level the concept of an inverse F -1 of many common functions Fis not used in a uniform way. For example, the definitions used for the inverse trigonometric functions may differ slightly depending on the choice of domain and/or branch cuts.
MathML adopts the view: if F is a function from a domain D to D', then the inverse G of F is a function over D' such that G(F(x)) = x for x in D. This definition does not assert that such an inverse exists for all or indeed any x in D, or that it is single-valued anywhere. Also, depending on the functions involved, additional properties such as F(G(y)) = y for y in D' may hold.
The
inverse element is applied to a function whenever an inverse is required. For example, application of the inverse sine function to
x, i.e. sin
-1 (x), is encoded as:
<apply> <apply> <inverse/> <sin/> </apply> <ci> x </ci> </apply>
While
arcsin is one of the predefined MathML functions, an explicit reference to sin
-1(x) might occur in a document discussing possible definitions of
arcsin.
Consider a document discussing the vectors
A = (a, b,
c) and
B = (d, e,
f), and later including the expression
V =
A +
B. It is important to be able to communicate the fact that wherever
A and
Bare used they represent a particular vector. The properties of that vector may determine aspects of operators such as
plus.
The simple fact that A is a vector can be communicated by using the markup
<ci type="vector">A</ci>
but this still does not communicate, for example, which vector is involved or its dimensions.
The declare construct is used to associate
specific properties or meanings with an object. The actual declaration
itself is not rendered visually (or in any other form). However, it
indirectly impacts the semantics of all affected uses of the declared
object.
The scope of a declaration is, by default, local to the MathML element
in which the declaration is made. If the scopeattribute of the declare
element is set to global, the declaration applies to
the entire MathML expression in which it appears.
The uses of the declare element range from
resetting default attribute values to associating an expression with a
particular instance of a more elaborate structure. Subsequent uses of the
original expression (within the scope of the declare) play the same semantic role as would the
paired object.
For example, the declaration
<declare>
<ci> A </ci>
<vector>
<ci> a </ci>
<ci> b </ci>
<ci> c </ci>
</vector>
</declare>
specifies that A stands for the particular vector (a,
b, c) so that subsequent uses of A as in
V = A + B can take this into account. When declare is used in this way, the actual encoding
<apply>
<eq/>
<ci> V </ci>
<apply>
<plus/>
<ci> A </ci>
<ci> B </ci>
</apply>
</apply>
remains unchanged but the expression can be interpreted properly as vector addition.
There is no requirement to declare an expression to stand for a specific object. For example, the declaration
<declare type="vector"> <ci> A </ci> </declare>
specifies that
A is a vector without indicating the number of components or the values of specific components. The possible values for the
type attribute include all the predefined container element names such as
vector,
matrix or
set (see
Section 4.3.2.9 [type]).
The lambda calculus allows a user to construct a function from a variable and an expression. For example, the lambda construct underlies the common mathematical idiom illustrated here:
Let f be the function taking x to x 2 + 2
There are various notations for this concept in mathematical literature, such as
(x,
F(x)) = F or
(x,
[F]) =F, where x is a free variable in F.
This concept is implemented in MathML with the lambda element. A lambda construct with n
internal variables is encoded by a lambda element
with n+1 children. All but the last child must be bvar elements containing the identifiers of the
internal variables. The last child is an expression defining the
function. This is typically an apply, but can also
be any container element.
The following constructs
(x, sin(x+1)):
<lambda>
<bvar><ci> x </ci></bvar>
<apply>
<sin/>
<apply>
<plus/>
<ci> x </ci>
<cn> 1 </cn>
</apply>
</apply>
</lambda>
To use
declare and
lambda to construct the function
f for which
f(
x) =
x
2 +
x + 3 use:
<declare type="fn">
<ci> f </ci>
<lambda>
<bvar><ci> x </ci></bvar>
<apply>
<plus/>
<apply>
<power/>
<ci> x </ci>
<cn> 2 </cn>
</apply>
<ci> x </ci>
<cn> 3 </cn>
</apply>
</lambda>
</declare>
The following markup declares and constructs the function J such that J(x, y) is the integral from x to y of t 4 with respect to t.
<declare type="fn">
<ci> J </ci>
<lambda>
<bvar><ci> x </ci></bvar>
<bvar><ci> y </ci></bvar>
<apply> <int/>
<bvar>
<ci> t </ci>
</bvar>
<lowlimit>
<ci> x </ci>
</lowlimit>
<uplimit>
<ci> y </ci>
</uplimit>
<apply> <power/>
<ci>t</ci>
<cn>4</cn>
</apply>
</apply>
</lambda>
</declare>
The function J can then in turn be applied to an argument pair.
The last example of the preceding section illustrates the use of
qualifier elements
lowlimit,
uplimit, and
bvar used in conjunction with the
int element. A number of common mathematical constructions involve additional data that is either implicit in conventional notation, such as a bound variable, or thought of as part of the operator rather than an argument, as is the case with the limits of a definite integral.
Content markup uses qualifier elements in conjunction with a number of operators, including integrals, sums, series, and certain differential operators. Qualifier elements appear in the same
apply element with one of these operators. In general, they must appear in a certain order, and their precise meaning depends on the operators being used. For details, see
Section 4.2.3.2 [Operators taking Qualifiers].
The qualifier element
bvar is also used in another important MathML construction. The
condition element is used to place conditions on bound variables in other expressions. This allows MathML to define sets by rule, rather than enumeration, for example. The following markup, for instance, encodes the set {
x |
x < 1}:
<set>
<bvar><ci> x </ci></bvar>
<condition>
<apply>
<lt/>
<ci> x </ci>
<cn> 1 </cn>
</apply>
</condition>
</set>
While the primary role of the MathML content element set is to directly encode the mathematical structure of expressions independent of the notation used to present the objects, rendering issues cannot be ignored. Each content element has a default rendering, given in Section 4.4 [The Content Markup Elements], and several mechanisms (including Section 4.3.3.2 [General Attributes]) are provided for associating a particular rendering with an object.
Containers provide a means for the construction of mathematical objects of a given type.
| Tokens |
ci,
cn,
csymbol |
| Constructors |
interval,
list,
matrix,
matrixrow,
set,
vector,
apply,
reln,
fn,
lambda |
| Specials |
declare |
Token elements are typically the leaves of the MathML expression tree. Token elements are used to indicate mathematical identifiers, numbers and symbols.
It is also possible for the canonically empty operator elements such as
exp,
sin and
cos to be leaves in an expression tree. The usage of operator elements is described in
Section 4.2.3 [Functions, Operators and Qualifiers].
cn element is the MathML token element used to represent numbers. The supported types of numbers include:
real,
integer,
rational,
complex-cartesian, and
complex-polar, with
real being the default type. An attribute
base (with default value
10) is used to help specify how the content is to be parsed. The content itself is essentially
PCDATA, separated by
<sep/> when two parts are needed in order to fully describe a number. For example, the real number 3 is constructed by
<cn type="real"> 3 </cn>, while the rational number 3/4 is constructed as
<cn type="rational"> 3<sep/>4 </cn>. The detailed structure and specifications are provided in
Section 4.4.1.1 [Number (cn)].ci element, or
`content identifier' is used to construct a variable, or an identifier. A
type attribute indicates the type of object the symbol represents. Typically,
ci represents a real scalar, but no default is specified. The content is either
PCDATA or a general presentation construct (see
Section 3.1.5 [Summary of Presentation Elements]). For example,
<ci> <msub> <mi>c</mi> <mn>1</mn> </msub> </ci>encodes an atomic symbol that displays visually as c 1which, for purposes of content, is treated as a single symbol representing a real number. The detailed structure and specifications is provided in Section 4.4.1.2 [Identifier (
ci)].csymbol element, or
`content symbol' is used to construct a symbol whose semantics are not part of the core content elements provided by MathML, but defined externally.
csymbol does not make any attempt to describe how to map the arguments occurring in any application of the function into a new MathML expression. Instead, it depends on its
definitionURL attribute to point to a particular meaning, and the
encoding attribute to give the syntax of this definition. The content of a
csymbol is either
PCDATAor a general presentation construct (see
Section 3.1.5 [Summary of Presentation Elements]). For example,
<csymbol definitionURL="www.vnbooks.com/ContDiffFuncs.htm"
encoding="text">
<msup>
<mi>C</mi>
<mn>2</mn>
</msup>
</csymbol>
encodes an atomic symbol that displays visually as
C
2 and that, for purposes of content, is treated as a single symbol representing the space of twice-differentiable continuous functions. The detailed structure and specifications is provided in
Section 4.4.1.3 [Externally defined symbol (csymbol)].
MathML provides a number of elements for combining elements into familiar compound objects. The compound objects include things like lists, sets. Each constructor produces a new type of object.
interval element is described in detail in
Section 4.4.2.4 [Interval (interval)]. It denotes an interval on the real line with the values represented by its children as end points. The
closure attribute is used to qualify the type of interval being represented. For example,
<interval closure="open-closed"> <ci> a </ci> <ci> b </ci> </interval>represents the open-closed interval often written (a, b].
set and
list elements are described in detail in
Section 4.4.6.1 [Set (set)] and
Section 4.4.6.2 [List (list)]. Typically, the child elements of a possibly empty
list element are the actual components of an ordered
list. For example, an ordered list of the three symbols
a,
b, and
c is encoded as
<list> <ci> a </ci> <ci> b </ci> <ci> c </ci> </list>Alternatively,
bvar and
condition elements can be used to define lists where membership depends on satisfying certain conditions.
An
order attribute, which is used to specify what ordering is to be used. When the nature of the child elements permits, the ordering defaults to a numeric or lexicographic ordering.
Sets are structured much the same as lists except that there is no implied ordering and the
type of set may be
normal or
multiset with
multiset indicating that repetitions are allowed.
For both sets and lists, the child elements must be valid MathML content elements. The type of the child elements is not restricted. For example, one might construct a list of equations, or inequalities.matrix element is used to represent mathematical matrices. It is described in detail in
Section 4.4.10.2 [Matrix (matrix)]. It has zero or more child elements, all of which are
matrixrow elements. These in turn expect zero or more child elements that evaluate to algebraic expressions or numbers. These sub-elements are often real numbers, or symbols as in
<matrix> <matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow> <matrixrow> <cn> 3 </cn> <cn> 4 </cn> </matrixrow> </matrix>The
matrixrow elements must always be contained inside of a matrix, and all rows in a given matrix must have the same number of elements.
Note that the behavior of the
matrix and
matrixrow elements is substantially different from the
mtable and
mtr presentation elements.vector element is described in detail in
Section 4.4.10.1 [Vector (vector)]. It constructs vectors from an
n-dimensional vector space so that its
n child elements typically represent real or complex valued scalars as in the three-element vector
<vector>
<apply>
<plus/>
<ci> x </ci>
<ci> y </ci>
</apply>
<cn> 3 </cn>
<cn> 7 </cn>
</vector>
apply element is described in detail in
Section 4.4.2.1 [Apply (apply)]. Its purpose is apply a function or operator to its arguments to produce an an expression representing an element of the range of the function. It is involved in everything from forming sums such as
a +
b as in
<apply> <plus/> <ci> a </ci> <ci> b </ci> </apply>through to using the sine function to construct sin(a) as in
<apply> <sin/> <ci> a </ci> </apply>or constructing integrals. Its usage in any particular setting is determined largely by the properties of the function (the first child element) and as such its detailed usage is covered together with the functions and operators in Section 4.2.3 [Functions, Operators and Qualifiers].
reln element is described in detail in
Section 4.4.2.2 [Relation (reln)]. It was used in MathML 1.0 to construct an expression such as
a =
b, as in
<reln><eq/> <ci> a </ci> <ci> b </ci> </reln>indicating an intended comparison between two mathematical values. MathML 2.0 takes the view that this should be regarded as the application of a boolean function, and as such could be constructed using
apply. The use of
reln with logical operators is supported
for reasons of backwards compatibility, but deprecated in favour of
apply.fn element was used in MathML 1.0 to make
explicit the fact that an expression is being used as a function or
operator. This is allowed in MathML 2.0 for backwards compatibility,
but is deprecated, as the use of
an expression as a function or operator is clear from its position as
the first child of an
apply.
fn is discussed in detail in
Section 4.4.2.3 [Function (fn)].lambda element is used to construct a user-defined function from an expression and one or more free variables. The lambda construct with
n internal variables takes
n+1 children. The first (second, up to
n) is a
bvar containing the identifiers of the internal variables. The last is an expression defining the function. This is typically an
apply, but can also be any container element. The following constructs
(x, sin
x)
<lambda>
<bvar><ci> x </ci></bvar>
<apply>
<sin/>
<ci> x </ci>
</apply>
</lambda>
The following constructs the constant function
(x, 3)
<lambda> <bvar><ci> x </ci></bvar> <cn> 3 </cn> </lambda>
The
declare construct is described in detail in
Section 4.4.2.8 [Declare (declare)]. It is special in that its entire purpose is to modify the semantics of other objects. It is not rendered visually or aurally.
The need for declarations arises any time a symbol (including more general presentations) is being used to represent an instance of an object of a particular type. For example, you may wish to declare that the symbolic identifier V represents a vector.
The declaration
<declare type="vector"><ci>V</ci></declare>
resets the default type attribute of
<ci>V</ci> to
vector for all affected occurrences of
<ci>V</ci>. This avoids having to write
<ci type="vector">V</ci> every time you use the symbol.
More generally,
declare can be used to associate expressions with specific content. For example, the declaration
<declare>
<ci>F</ci>
<lambda>
<bvar><ci> U </ci></bvar>
<apply>
<int/>
<bvar><ci> x </ci></bvar>
<lowlimit><cn> 0 </cn></lowlimit>
<uplimit><ci> a </ci></uplimit>
<ci> U </ci>
</apply>
</lambda>
</declare>
associates the symbol
F with a new function defined by the
lambda construct. Within the scope where the declaration is in effect, the expression
<apply> <ci>F</ci> <ci> U </ci> </apply>
stands for the integral of U from 0 to a.
The
declare element can also be used to change the definition of a function or operator. For example, if the URL
http://.../MathML:noncommutplus described a non-commutative plus operation encoded in Maple syntax, then the declaration
<declare definitionURL="http://.../MathML:noncommutplus"
encoding="Maple">
<plus/>
</declare>
would indicate that all affected uses of
plus are to be interpreted as having that definition of
plus.
The operators and functions defined by MathML can be divided into categories as shown in the table below.
| unary arithmetic |
exp,
factorial,
minus,
abs,
conjugate,
arg,
real,
imaginary |
| unary logical |
not |
| unary functional |
inverse,
ident |
| unary elementary classical functions |
sin,
cos,
tan,
sec,
csc,
cot,
sinh,
cosh,
tanh,
sech,
csch,
coth,
arcsin,
arccos,
arctan,
arccosh,
arccot,
arccoth,
arccsc,
arccsch,
arcsec,
arcsech,
arcsinh,
arctanh,
exp,
ln,
log |
| unary linear algebra |
determinant,
transpose |
| unary calculus and vector calculus |
divergence,
grad,
curl,
laplacian |
| unary set-theoretic |
card |
| binary arithmetic |
quotient,
divide,
minus,
power,
rem |
| binary logical |
implies,
equivalent,
approx |
| binary set operators |
setdiff |
| binary linear algebra |
vectorproduct,
scalarproduct,
outerproduct |
| n-ary arithmetic |
plus,
times,
max,
min,
gcd,
lcm
|
| n-ary statistical |
mean,
sdev,
variance,
median,
mode |
| n-ary logical |
and,
or,
xor |
| n-ary linear algebra |
selector |
| n-ary set operator |
union,
intersect |
| n-ary functional |
fn,
compose |
| integral, sum, product operators |
int,
sum,
product |
| differential operator |
diff,
partialdiff |
| quantifier |
forall,
exists |
From the point of view of usage, MathML regards functions (for example
sin and
cos) and operators (for example
plus and
times) in the same way. MathML predefined functions and operators are all canonically empty elements.
Note that the
csymbol element can be used to construct a user-defined symbol that can be used as a function or operator.
MathML functions can be used in two ways. They can be used as the operator within an
apply element, in which case they refer to a function evaluated at a specific value. For example,
<apply> <sin/> <cn>5</cn> </apply>
denotes a real number, namely sin(5).
MathML functions can also be used as arguments to other operators, for example
<apply> <plus/><sin/><cos/> </apply>
denotes a function, namely the result of adding the sine and cosine functions in some function space. (The default semantic definition of
plus is such that it infers what kind of operation is intended from the type of its arguments.)
The number of child elements in the
apply is defined by the element in the first (i.e. operator) position.
Unary operators are followed by exactly one other child element within the
apply.
Binary operators are followed by exactly two child elements.
N-ary operators are followed by zero or more child elements.
The one exception to these rules is that
declare elements may be inserted in any position except the first.
declare elements are not counted when satisfying the child element count for an
applycontaining a unary or binary operator element.
Integral, sum, product and differential operators are discussed below in Section 4.2.3.2 [Operators taking Qualifiers].
The table below contains the qualifiers and the operators taking qualifiers in MathML.
| qualifiers |
lowlimit,
uplimit,
bvar,
degree,
logbase,
interval,
condition |
| operators |
int,
sum,
product,
root,
diff,
partialdiff,
limit,
log,
moment,
min,
max,
forall,
exists |
Operators taking qualifiers are canonically empty functions that differ from ordinary empty functions only in that they support the use of special
qualifier elements to specify their meaning more fully. They are used in exactly the same way as ordinary operators, except that when they are used as operators, certain qualifier elements are also permitted to be in the enclosing
apply. They always precede the argument if it is present. If more than one qualifier is present, they appear in the order
bvar,
lowlimit,
uplimit,
interval,
condition,
degree,
logbase. A typical example is:
<apply>
<int/>
<bvar><ci>x</ci></bvar>
<lowlimit><cn>0</cn></lowlimit>
<uplimit><cn>1</cn></uplimit>
<apply>
<power/>
<ci>x</ci>
<cn>2</cn>
</apply>
</apply>
It is also valid to use qualifier schema with a function not applied to an argument. For example, a function acting on integrable functions on the interval [0,1] might be denoted:
<fn>
<apply>
<int/>
<bvar><ci>x</ci></bvar>
<lowlimit><cn>0</cn></lowlimit>
<uplimit><cn>1</cn></uplimit>
</apply>
</fn>
The meaning and usage of qualifier schema varies from function to function. The following list summarizes the usage of qualifier schema with the MathML functions taking qualifiers.
int function accepts the
lowlimit,
uplimit,
bvar,
interval and
condition schemata. If both
lowlimit and
uplimit schema are present, they denote the limits of a definite integral. The domain of integration may alternatively be specified using interval or condition. The
bvar schema signifies the variable of integration. When used with
int, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.diff function accepts the
bvar schema. The
bvar schema specifies with respect to which variable the derivative is being taken. The
bvar may itself contain a
degree schema that is used to specify the order of the derivative, i.e. a first derivative, a second derivative, etc. For example, the second derivative of
f with respect to
x is:
<apply>
<diff/>
<bvar>
<ci> x </ci>
<degree>
<cn> 2 </cn>
</degree>
</bvar>
<apply><fn><ci>f</ci></fn>
<ci> x </ci>
</apply>
</apply>
partialdiff function accepts zero or more
bvar schemata. The
bvarschema specify with respect to which variables the derivative is being taken. The
bvar elements may themselves contain
degree schemata that are used to specify the order of the derivative. Variables specified by multiple
bvar elements will be used in order as the variable of differentiation in mixed partials. When used with
partialdiff, the
degree schema is expected to contain a single child schema. For example,
<apply> <partialdiff/> <bvar><ci>x</ci></bvar> <bvar><ci>y</ci></bvar> <fn><ci>f</ci></fn> </apply>denote the mixed partial (d 2 / d xd y) f.
sum and
productfunctions accept the
bvar,
lowlimit,
uplimit,
interval and
conditionschemata. If both
lowlimit and
uplimit schemata are present, they denote the limits of the sum or product. The limits may alternatively be specified using the
interval or
conditionschema. The
bvar schema signifies the index variable in the sum or product. A typical example might be:
<apply>
<sum/>
<bvar><ci>i</ci></bvar>
<lowlimit><cn>0</cn></lowlimit>
<uplimit><cn>100</cn></uplimit>
<apply>
<power/>
<ci>x</ci>
<ci>i</ci>
</apply>
</apply>
When used with
sum or
product, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.limit function accepts zero or more
bvar schemata, and optional
condition and
lowlimitschemata. A
condition may be used to place constraints on the
bvar. The
bvar schema denotes the variable with respect to which the limit is being taken. The
lowlimit schema denotes the limit point. When used with
limit, the
bvar and
lowlimit schemata are expected to contain a single child schema; otherwise an error is generated.log function accepts only the
logbase schema. If present, the
logbase schema denotes the base with respect to which the logarithm is being taken. Otherwise, the log is assumed to be base 10. When used with
log, the
logbase schema is expected to contain a single child schema; otherwise an error is generated.moment function accepts only the
degree schema. If present, the
degree schema denotes the order of the moment. Otherwise, the moment is assumed to be the first order moment. When used with
moment, the
degree schema is expected to contain a single child schema; otherwise an error is generated.min and
maxfunctions accept a
bvar schema in cases where the maximum or minimum is being taken over a set of values specified by a
condition schema together with an expression to be evaluated on that set.
bvar element was optional when using a
condition; if a
condition element containing a single variable was given by itself following a
min or
max operator, the variable was implicitly
assumed to be bound, and the expression to be maximized or minimized
(if absent) was assumed to be the single bound variable. This usage
is deprecated in MathML 2.0 in
favour of explicitly stating the bound variable(s) and the expression
to be maximised in all cases.
The
min and
max elements may also be applied to a list of values in which case no qualifier schemata are used. For examples of all three usages, see
Section 4.4.3.4 [Maximum and minimum (max,
min)].forall and
exists are used in conjuction with one or more
bvar schemata to represent simple logical assertions. There are two ways of using the logical quantifier operators. The first usage is for representing a simple, quantified assertion. For example, the statement
`there exists
x< 9' would be represented as:
<apply>
<exists/>
<bvar><ci> x </ci></bvar>
<apply><lt/>
<ci> x </ci><cn> 9 </cn>
</apply>
</apply>
The second usage is for representing implications. Hypotheses are given by a
condition element following the bound variables. For example the statement
`for all
x < 9,
x < 10' would be represented as:
<apply>
<forall/>
<bvar><ci> x </ci></bvar>
<condition>
<apply><lt/>
<ci> x </ci><cn> 9 </cn>
</apply>
</condition>
<apply><lt/>
<ci> x </ci><cn> 10 </cn>
</apply>
</apply>
Note that in both usages one or more
bvar qualifiers are mandatory.
| binary relation |
neq,
equivalent,
approx |
| binary logical relation |
implies |
| binary set relation |
in,
notin,
notsubset,
notprsubset |
| binary series relation |
tendsto |
| n-ary relation |
eq,
leq,
lt,
geq,
gt |
| n-ary set relation |
subset,
prsubset |
The MathML content tags include a number of canonically empty elements which denote arithmetic and logical relations. Relations are characterized by the fact that, if an external application were to evaluate them (MathML does not specify how to evaluate expressions), they would typically return a truth value. By contrast, operators generally return a value of the same type as the operands. For example, the result of evaluating a < b is either true or false (by contrast, 1 + 2 is again a number).
Relations are bracketed with their arguments using the
apply element in the same way as other functions. In MathML 1.0, relational operators were bracketed using
reln. This usage, although still supported,
is now deprecated in favour of
apply. The element for the relational operator is the first child element of the
apply. Thus, the example from the preceding paragraph is properly marked up as:
<apply> <lt/> <ci>a</ci> <ci>b</ci> </apply>
It is an error to enclose a relation in an element other than
apply or
reln.
The number of child elements in the
apply is defined by the element in the first (i.e. relation) position.
Unary relations are followed by exactly one other child element within the
apply.
Binary relations are followed by exactly two child elements.
N-ary relations are followed by zero or more child elements.
The one exception to these rules is that
declare elements may be inserted in any position except the first.
declare elements are not counted when satisfying the child element count for an
applycontaining a unary or binary relation element.
| condition |
condition |
The
condition element is used to define the
`such that' construct in mathematical expressions. Condition elements are used in a number of contexts in MathML. They are used to construct objects like sets and lists by rule instead of by enumeration. They can be used with the
forall and
exists operators to form logical expressions. And finally, they can be used in various ways in conjunction with certain operators. For example, they can be used with and
int element to specify domains of integration, or to specify argument lists for operators like
min and
max.
The
condition element is always used together with one or more
bvar elements.
The exact interpretation depends on the context, but generally speaking, the
condition element is used to restrict the permissible values of a bound variable appearing in another expression to those that satisfy the relations contained in the
condition. Similarly, when the
condition element contains a
set, the values of the bound variables are restricted to that set.
A condition element contains a single child that is either a
apply, or a
reln element (deprecated). Compound conditions are
indicated by applying relations such as
and inside the child of the condition.
The following encodes `there exists x such that x 5 < 3'.
<apply>
<exists/>
<bvar><ci> x </ci></bvar>
<condition>
<apply><lt/>
<apply>
<power/>
<ci>x</ci>
<cn>5</cn>
</apply>
<cn>3</cn>
</apply>
</condition>
</apply>
The next example encodes `for all xin N there exists prime numbers p, q such that p+ q= 2x'.
<apply>
<forall/>
<bvar><ci>x</ci></bvar>
<condition>
<apply><in/>
<ci>x</ci>
<csymbol encoding="text" definitionURL="www.naturalnums.htm">N</csymbol>
</apply>
</condition>
<apply><exists/>
<bvar><ci>p</ci></bvar>
<bvar><ci>q</ci></bvar>
<condition>
<apply><and/>
<apply><in/><ci>p</ci>
<csymbol encoding="text" definitionURL="www.primes.htm">P</csymbol>
</apply>
<apply><in/><ci>q</ci>
<csymbol encoding="text" definitionURL="www.primes.htm">P</csymbol>
</apply>
<apply><eq/>
<apply><plus/><ci>p</ci><ci>q</ci></apply>
<apply><times/><cn>2</cn><ci>x</ci></apply>
</apply>
</apply>
</condition>
</apply>
</apply>
A third example shows the use of quantifiers with
condition. The following markup encodes
`there exists
x < 3 such that
x
2 = 4'.
<apply>
<exists/>
<bvar><ci> x </ci></bvar>
<condition>
<apply><lt/><ci>x</ci><cn>3</cn></apply>
</condition>
<apply>
<eq/>
<apply>
<power/><ci>x</ci><cn>2</cn>
</apply>
<cn>4</cn>
</apply>
</apply>
| mappings |
semantics,
annotation,
annotation-xml |
The use of content markup rather than presentation markup for mathematics is sometimes referred to as semantic tagging [Buswell1996]. The parse-tree of a valid element structure using MathML content elements corresponds directly to the expression tree of the underlying mathematical expression. We therefore regard the content tagging itself as encoding the syntax of the mathematical expression. This is, in general, sufficient to obtain some rendering and even some symbolic manipulation (e.g. polynomial factorization).
However, even in such apparently simple expressions as
X +
Y, some additional information may be required for applications such as computer algebra. Are
X and
Y integers, or functions, etc.?
`Plus' represents addition over which field? This additional information is referred to as
semantic mapping. In MathML, this mapping is provided by the
semantics,
annotation and
annotation-xml elements.
The
semantics element is the container element for the MathML expression together with its semantic mappings.
semantics expects a variable number of child elements. The first is the element (which may itself be a complex element structure) for which this additional semantic information is being defined. The second and subsequent children, if any, are instances of the elements
annotation and/or
annotation-xml.
The
semantics tags also accepts the
definitionURL and
encoding attributes for use by external processing applications. One use might be a URI for a semantic content dictionary, for example. Since the semantic mapping information might in some cases be provided entirely by the
definitionURLattribute, the
annotation or
annotation-xml elements are optional.
The
annotation element is a container for arbitrary data. This data may be in the form of text, computer algebra encodings, C programs, or whatever a processing application expects.
annotation has an attribute
encoding defining the form in use. Note that the content model of
annotation is
PCDATA, so care must be taken that the particular encoding does not conflict with XML parsing rules.
The
annotation-xml element is a container for semantic information in well-formed XML. For example, an XML form of the OpenMath semantics could be given. Another possible use here is to embed, for example, the presentation tag form of a construct given in content tag form in the first child element of
semantics (or vice versa).
annotation-xml has an attribute
encoding defining the form in use.
For example:
<semantics>
<apply>
<divide/>
<cn>123</cn>
<cn>456</cn>
</apply>
<annotation encoding="Mathematica">
N[123/456, 39]
</annotation>
<annotation encoding="TeX">
$0.269736842105263157894736842105263157894\ldots$
</annotation>
<annotation encoding="Maple">
evalf(123/456, 39);
</annotation>
<annotation-xml encoding="MathML-Presentation">
<mrow>
<mn> 0.269736842105263157894 </mn>
<mover accent='true'>
<mn> 736842105263157894 </mn>
<mo> ‾ </mo>
</mover>
</mrow>
</annotation-xml>
<annotation-xml encoding="OpenMath">
<OMA>...</OMA>
</annotation-xml>
</semantics>
where
OMA is the element defining the additional semantic information.
Of course, providing an explicit semantic mapping at all is optional, and in general would only be provided where there is some requirement to process or manipulate the underlying mathematics.
Although semantic mappings can easily be provided by various proprietary, or highly specialized encodings, there are no widely available, non-proprietary standard schemes for semantic mapping. In part to address this need, the goal of the OpenMath effort is to provide a platform-independent, vendor-neutral standard for the exchange of mathematical objects between applications. Such mathematical objects include semantic mapping information. The OpenMath group has defined an SGML syntax for the encoding of this information
[OpenMath1996]. This element set could provide the basis of one
annotation-xml element set.
An attractive side of this mechanism is that the OpenMath syntax is specified in XML, so that a MathML expression together with its semantic annotations can be validated using XML parsers.
MathML provdies a collection of predefined constants and symbols which represent frequently-encountered concepts in K-12 mathematics. These include symbols for well-known sets, such as integers
integers and
rationals, and also some widely known constant symbols such as
false,
true,
exponentiale.
MathML functions, operators and relations can all be thought of as mathematical functions if viewed in a sufficiently abstract way. For example, the standard addition operator can be regarded as a function mapping pairs of real numbers to real numbers. Similarly, a relation can be thought of as a function from some space of ordered pairs into the set of values {true, false}. To be mathematically meaningful, the domain and range of a function must be precisely specified. In practical terms, this means that functions only make sense when applied to certain kinds of operands. For example, thinking of the standard addition operator, it makes no sense to speak of `adding' a set to a function. Since MathML content markup seeks to encode mathematical expressions in a way that can be unambiguously evaluated, it is no surprise that the types of operands is an issue.
MathML specifies the types of arguments in two ways. The first way is by providing precise instructions for processing applications about the kinds of arguments expected by the MathML content elements denoting functions, operators and relations. These operand types are defined in a dictionary of default semantic bindings for content elements, which is given in
Appendix C [Content Element Definitions]. For example, the MathML content dictionary specifies that for real scalar arguments the plus operator is the standard commutative addition operator over a field. The elements
cn has a
type attribute with a default value of
real. Thus some processors will be able to use this information to verify the validity of the indicated operations.
Although MathML specifies the types of arguments for functions, operators and relations, and provides a mechanism for typing arguments, a MathML-compliant processor is not required to do any type checking. In other words, a MathML processor will not generate errors if argument types are incorrect. If the processor is a computer algebra system, it may be unable to evaluate an expression, but no MathML error is generated.
Content element attributes are all of the type
CDATA, that is, any character string will be accepted as valid. In addition, each attribute has a list of predefined values, which a content processor is expected to recognize and process. The reason that the attribute values are not formally restricted to the list of predefined values is to allow for extension. A processor encountering a value (not in the predefined list) which it does not recognize may validly process it as the default value for that attribute.
Each attribute is followed by the elements to which it can be applied.
base
10
closure
open,
closed,
open-closed,
closed-open.
The default value is
closed
definitionURL
definitionURL attribute would be some standard, machine-readable format. However, there are several reasons why MathML does not require such a format.
First, no such format currently exists. There are several projects underway to develop and implement standard semantic encoding formats, most notably the OpenMath effort. But by nature, the development of a comprehensive system of semantic encoding is a very large enterprise, and while much work has been done, much additional work remains. Therefore, even though the
definitionURL is designed and intended for use with a formal semantic encoding language such as OpenMath, it is premature to require any one particular format.
Another reason for leaving the format of the
definitionURL attribute unspecified is that there will always be situations where some non-standard format is preferable. This is particularly true in situations where authors are describing new ideas.
It is anticipated that in the near term, there will be a variety of renderer-dependent implementations of the
definitionURL attribute. For example, a translation tool might simply prompt the user with the specified definition in situations where the proper semantics have been overridden, and in this case, human-readable definitions will be most useful. Other software may utilize OpenMath encodings. Still other software may use proprietary encodings, or look for definitions in any of several formats.
As a consequence, authors need to be aware that there is no guarantee a generic renderer will be able to take advantage of information pointed to by the
definitionURL attribute. Of course, when widely-accepted standardized semantic encodings are available, the definitions pointed to can be replaced without modifying the original document. However, this is likely to be labor intensive.
There is no default value for the
definitionURLattribute, i.e. the semantics are defined within the MathML fragment, and/or by the MathML default semantics.encoding
csymbol ,
semantics and operator elements, the syntax of the target referred to by
definitionURL. Predefined values are
MathML-Presentation,
MathML-Content. Other typical values:
TeX,
OpenMath.
The default value is "", i.e. unspecified.nargs
nary, or any numeric string.
The default value is
1
occurrence
prefix,
infix,
function-model.
The default value is
function-model
order
lexicographic,
numeric.
The default value is
numeric
scope
local,
global.
local means the containing MathML element.global means the containing
math element.local.
At present, declarations cannot affect anything outside of the containing
math element. Ideally, one would like to make document-wide declarations by setting the value of the
scope attribute to be
global-document. However, the proper mechanism for document-wide declarations very much depends on details of the way in which XML will be embedded in HTML, future XML style sheet mechanisms, and the underlying Document Object Model.
Since these supporting technologies are still in flux at present, the MathML specification does not include
global-document as a pre-defined value of the
scope attribute. It is anticipated, however, that this issue will be revisited in future revisions of MathML as supporting technologies stabilize. In the near term, MathML implementors that wish to simulate the effect of a document-wide declaration are encouraged to pre-process documents in order to distribute document-wide declarations to each individual
math element in the document.type
integer,
rational,
real,
float,
complex,
complex-polar,
complex-cartesian,
constant.
The default value is
real.
Notes. Each data type implies that the data adheres to certain formating conventions, detailed below. If the data fails to conform to the expected format, an error is generated. Details of the individual formats are:
base is specified, then the digits are interpreted as being digits computed to that base.
A real number may also be presented in scientific notation. Such numbers have two parts (a mantissa and an exponent) separated by
`e'. The first part is a real number, while the second part is an integer exponent indicating a power of the base. For example, 12.3e5 represents 12.3 times 10^5.baseattribute. If
base is present, it specifies the base for the digit encoding, and it specifies it base ten. Thus
base='16' specifies a hex encoding. When
base > 10, letters are added in alphabetical order as digits. The legitimate values for
base are therefore between 2 and 36.<sep/>. If
base is present, it specifies the base used for the digit encoding of both integers.<sep/>.<sep/>.constant type is used to denote named constants. For example, an instance of
<cn type="constant">π</cn>should be interpreted as having the semantics of the mathematical constant Pi. The data for a constant
cn tag may be one of the following common constants:
| Symbol | Value |
π |
The usual
π of trigonometry: approximately 3.141592653... |
ⅇ (or
ⅇ) |
The base for natural logarithms: approximately 2.718281828 ... |
ⅈ (or
ⅈ) |
Square root of -1 |
γ |
Euler's constant: approximately 0.5772156649... |
∞ (or
&infty;) |
Infinity. Proper interpretation varies with context |
&true; |
the logical constant
true |
&false; |
the logical constant
false |
&NotANumber; (or
&NaN;) |
represents the result of an ill-defined floating point division |
integer,
rational,
real,
float,
complex,
complex-polar,
complex-cartesian,
constant, or the name of any content element. The meaning of the various attribute values is the same as that listed above for the
cn element.
The default value is "", i.e. unspecified.ci , i.e. a generic identifiernormal,
multiset.
multiset indicates that repetitions are allowed.
The default value is
normal.above,
below,
two-sided.
The default value is
above.type
The
type attribute, in addition to conveying semantic information, can be interpreted to provide rendering information. For example in
<ci type="vector">V</ci>
a renderer could display a bold V for the vector.
All content elements support the following general attributes that can be used to modify the rendering of the markup.
class
style
id
other
The
class,
style and
id attributes are intended for compatibility with Cascading Style Sheets (CSS), as described in
Section 2.3.5 [Attributes Shared by all MathML Elements].
Content or semantic tagging goes along with the (frequently implicit) premise that, if you know the semantics, you can always work out a presentation form. When an author's main goal is to mark up re-usable, evaluatable mathematical expressions, the exact rendering of the expression is probably not critical, provided that it is easily understandable. However, when an author's goal is more along the lines of providing enough additional semantic information to make a document more accessible by facilitating better visual rendering, voice rendering, or specialized processing, controlling the exact notation used becomes more of an issue.
MathML elements accept an attribute
other (see
Section 7.2.3 [Attributes for unspecified data]), which can be used to specify things not specifically documented in MathML. On content tags, this attribute can be used by an author to express a
preference between equivalent forms for a particular content element construct, where the selection of the presentation has nothing to do with the semantics. Examples might be
Thus, if a particular renderer recognized a display attribute to select between script-style and display-style fractions, an author might write
<apply other='display="scriptstyle"'> <divide/> <mn> 1 </mn> <mi> x </mi> </apply>
to indicate that the rendering 1/ x is preferred.
The information provided in the
other attribute is intended for use by specific renderers or processors, and therefore, the permitted values are determined by the renderer being used. It is legal for a renderer to ignore this information. This might be intentional, in the case of a publisher imposing a house style, or simply because the renderer does not understand them, or is unable to carry them out.
This section provides detailed descriptions of the MathML content tags. They are grouped in categories that broadly reflect the area of mathematics from which they come, and also the grouping in the MathML DTD. There is no linguistic difference in MathML between operators and functions. Their separation here and in the DTD is for reasons of historical usage.
When working with the content elements, it can be useful to keep in mind the following.
The available content elements are:
vector
matrix
matrixrow
determinant
transpose
selector
vectorproduct (MathML 2.0)scalarproduct (MathML 2.0)outerproduct (MathML 2.0)integers (MathML2.0)reals (MathML2.0)rationals (MathML2.0)naturalnumbers (MathML2.0)complexes (MathML2.0)primes (MathML2.0)exponentiale (MathML2.0)imaginaryi (MathML2.0)notanumber (MathML2.0)true (MathML2.0)false (MathML2.0)emptyset (MathML2.0)pi (MathML2.0)eulergamma (MathML2.0)infinity (MathML2.0)
cn)The cn element is used to specify actual
numerical constants. The content model must provide sufficient information
that a number may be entered as data into a computational system. By
default, it represents a signed real number in base 10. Thus, the content
normally consists of PCDATA restricted to a sign, a string of
decimal digits and possibly a decimal point, or alternatively one of the
predefined symbolic constants such as π.
The cn element uses the attribute type to represent other types of numbers such as, for
example, integer, rational, real or complex, and uses the attribute base to specify the numerical base.
In addition to simple PCDATA, cn
accepts as content PCDATA separated by the (empty) element sep. This determines the different parts needed to
construct a rational or complex-cartesian number.
The cn element may also contain arbitrary
presentation markup in its content (see Chapter 3 [Presentation Markup]) so that its
presentation can be very elaborate.
Alternative input notations for numbers are possible, but must be
explicitly defined by using the definitionURL and
encoding attributes, to refer to a written
specification of how a sequence of real numbers separated by <sep/> should be interpreted.
All attributes are CDATA:
real,
integer,
rational,
complex-cartesian,
complex-polar,
constant
CDATA for XML DTD) between 2 and 36.
<cn type="real"> 12345.7 </cn> <cn type="integer"> 12345 </cn> <cn type="integer" base="16"> AB3 </cn> <cn type="rational"> 12342 <sep/> 2342342 </cn> <cn type="complex-cartesian"> 12.3 <sep/> 5 </cn> <cn type="complex-polar"> 2 <sep/> 3.1415 </cn> <cn type="constant"> π </cn>
By default, a contiguous block of
PCDATA contained in a
cn element should render as if it were wrapped in an
mn presentation element. Similarly, presentation markup contained in a
cn element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were contained wrapped in an
mrow element, with contiguous blocks of
PCDATAwrapped in
mn elements.
However, not all mathematical systems that encounter content based tagging do visual or aural rendering. The receiving applications are free to make use of a number in the manner it normally handles numerical data. Some systems might simplify the rational number 12342/2342342 to 6171/1171171 while pure floating point based systems might approximate this as 0.5269085385e-2. All numbers might be re-expressed in base 10. The role of MathML is simply to record enough information about the mathematical object and its structure so that it may be properly parsed.
The following renderings of the above MathML expressions are included both to help clarify the meaning of the corresponding MathML encoding and as suggestions for authors of rendering applications. In each case, no mathematical evaluation is intended or implied.
ci)The
ci element is used to name an identifier in a MathML expression (for example a variable). Such names are used to identify mathematical objects. By default they are assumed to represent complex scalars. The
ci element may contain arbitrary presentation markup in its content (see
Chapter 3 [Presentation Markup]) so that its presentation as a symbol can be very elaborate.
The
ci element uses the
type attribute to specify the type of object that it represents. Valid types include
integer,
rational,
real,
float,
complex,
constant, and more generally, any of the names of the MathML container elements (e.g.
vector) or their type values. The
definitionURL and
encoding attributes can be used to extend the definition of
ci to include other types. For example, a more advanced use might require a
complex-vector.
<ci> x </ci>
<ci type="vector"> V </ci>
<ci>
<msub>
<mi>x</mi>
<mi>a</mi>
</msub>
</ci>
If the content of a
ci element is tagged using presentation tags, that presentation is used. If no such tagging is supplied then the
PCDATA content would typically be rendered as if it were the content of an
mi element. A renderer may wish to make use of the value of the type attribute to improve on this. For example, a symbol of type
vector might be rendered using a bold face. Typical renderings of the above symbols are:
csymbol)The
csymbol element allows a writer to create an element in MathML whose semantics are externally defined (i.e. not in the core MathML content). The element can then be used in a MathML expression as for example an operator or constant. Attributes are used to give the syntax and location of the external definition of the symbol semantics.
Use of
csymbol for referencing external semantics can be contrasted with use of the
semantics to attach additional information in-line (ie. within the MathML fragment) to a MathML construct. See
Section 4.2.6 [Syntax and Semantics]
All attributes are
CDATA:
definitionURL. This syntax might be text, or a formal syntax such as OpenMath.
<!-- reference to OpenMath formal syntax definition of Bessel function -->
<apply>
<csymbol encoding="OpenMath"
definitionURL="www.openmath.org/cds/BesselFunctions.ocd">
<msub><mi>J</mi><mn>0</mn></msub>
</csymbol>
<ci>y</ci>
</apply>
<!-- reference to human readable text description of Boltzmann's constant -->
<csymbol encoding="text"
definitionURL="www.uni.edu/universalconstants/Boltzmann.htm">
k
</csymbol>
By default, a contiguous block of
PCDATA contained in a
csymbol element should render as if it were wrapped in an
mo presentation element. Similarly, presentation markup contained in a
csymbol element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were contained wrapped in an
mrowelement, with contiguous blocks of
PCDATA wrapped in
mo elements. The examples above would render by default as
As
csymbol is used to support reference to externally defined semantics, it is a MathML error to have embedded content MathML elements within the
csymbolelement.
apply)The
apply element allows a function or operator to be applied to its arguments. Nearly all expression construction in MathML content markup is carried out by applying operators or functions to arguments. The first child of
apply is the operator, to be applied, with the other child elements as arguments.
The
apply element is conceptually necessary in order to distinguish between a function or operator, and an instance of its use. The expression constructed by applying a function to 0 or more arguments is always an element from the range of the function.
Proper usage depends on the operator that is being applied. For example, the
plus operator may have zero or more arguments. while the
minus operator requires one or two arguments to be properly formed.
If the object being applied as a function is not already one of the elements known to be a function (such as
fn,
sin or
plus) then it is treated as if it were the contents of an
fn element.
Some operators such as
diff and
int make use of
`named' arguments. These special arguments are elements that appear as children of the
apply element and identify
`parameters'such as the variable of differentiation or the domain of integration. These elements are discussed further in
Section 4.2.3.2 [Operators taking Qualifiers].
<apply> <factorial/> <cn>3</cn> </apply>
<apply> <plus/> <cn>3</cn> <cn>4</cn> </apply>
<apply> <sin/> <ci>x</ci> </apply>
A mathematical system that has been passed an
apply element is free to do with it whatever it normally does with such mathematical data. It may be that no rendering is involved (e.g. a syntax validator), or that the
`function application' is evaluated and that only the result is rendered (e.g. sin(0)
0).
When an unevaluated
`function application' is rendered there are a wide variety of appropriate renderings. The choice often depends on the function or operator being applied. Applications of basic operations such as
plus are generally presented using an infix notation while applications of
sinwould use a more traditional functional notation such as sin(x). Consult the default rendering for the operator being applied.
Applications of user-defined functions (see
csymbol,
fn) that are not evaluated by the receiving or rendering application would typically render using a traditional functional notation unless an alternative presentation is specified using the
semantics tag.
reln)The
reln element was used in MathML 1.0 to construct an equation or relation. Relations were constructed in a manner exactly analogous to the use of
apply. This usage is deprecated in MathML 2.0 in favour of the more generally usable
apply.
The first child of
reln is the relational operator, to be applied, with the other child elements acting as arguments. See
Section 4.2.4 [Relations] for further details.
<reln> <eq/> <ci> a </ci> <ci> b </ci> </reln>
<reln> <lt/> <ci> a </ci> <ci> b </ci> </reln>
fn)The
fn element makes explicit the fact that a more general (possibly constructed) MathML object is being used in the same manner as if it were a pre-defined function such as
sin or
plus.
fn has exactly one child element, used to give the name (or presentation form) of the function. When
fn is used as the first child of an apply, the number of following arguments is determined by the contents of the
fn.
In MathML 1.0,
fn was also the primary mechanism used to extend the collection of
`known' mathematical functions. This usage is now deprecated in favour of the more generally applicable
csymbol element. (New functions may also be introduced by using
declare in conjunction with a
lambda expression.)
<fn><ci> L </ci> </fn>
<apply>
<fn>
<apply>
<plus/>
<ci> f </ci>
<ci> g </ci>
</apply>
</fn>
<ci>z</ci>
</apply>
An
fn object is rendered in the same way as its content. A rendering application may add additional adornments such as parentheses to clarify the meaning.
interval)The
interval element is used to represent simple mathematical intervals of the real number line. It takes an attribute
closure, which can take on any of the values
open,
closed,
open-closed, or
closed-open, with a default value of
closed.
More general domains are constructed by using the
condition and
bvar elements to bind free variables to constraints.
The
interval element expects
eithertwo child elements that evaluate to real numbers
or one child element that is a
condition defining the
interval.
<interval> <ci> a </ci> <ci> b </ci> </interval>
<interval closure="open-closed"> <ci> a </ci> <ci> b </ci> </interval>
![[a, b]](http://www.w3.org/TR/2000/WD-MathML2-20000328/image/f4014.gif)
![(a, b]](http://www.w3.org/TR/2000/WD-MathML2-20000328/image/f4015.gif)
inverse)The
inverse element is applied to a function in order to construct a generic expression for the functional inverse of that function. (See also the discussion of
inverse in
Section 4.2.1.5 [The inverse construct]). As with other MathML functions,
inverse may either be applied to arguments, or it may appear alone, in which case it represents an abstract inversion operator acting on other functions.
A typical use of the
inverse element is in an HTML document discussing a number of alternative definitions for a particular function so that there is a need to write and define
f
(-1)(x). To associate a particular definition with
f
(-1), use the
definitionURL and
encodingattributes.
<apply> <inverse/> <ci> f </ci> </apply>
<apply> <inverse definitionURL="../MyDefinition.htm" encoding="text"/> <ci> f </ci> </apply>
<apply>
<apply><inverse/>
<ci type="matrix"> a </ci>
</apply>
<ci> A </ci>
</apply>
The default rendering for a functional inverse makes use of a parenthesized exponent as in f (-1)(x).
sep)The sep element is to separate PCDATA
into separate tokens for parsing the contents of the various specialized
forms of the cn elements. For example, sep is used when specifying the real and imaginary
parts of a complex number (see Section 4.4.1 [Token Elements]). If it
occurs between MathML elements, it is a MathML error.
<cn type="complex"> 3 <sep/> 4 </cn>
The sep element is not directly rendered (see
Section 4.4.1 [Token Elements]).
condition)The condition element is used to place a
condition on one or more free variables or identifiers. The conditions may
be specified in terms of relations that are to be satisfied by the
variables, including general relationships such as set membership.
It is used to define general sets and lists in situations where the
elements cannot be explicitly enumerated. Condition contains either a
single apply or relnelement; the apply element
is used to construct compound conditions. For example, it is used below to
describe the set of all x such that x < 5. See the
discussion on sets in Section 4.4.6 [Theory of Sets]. See Section 4.2.5 [Conditions] for further details.
<condition> <apply><in/><ci> x </ci><ci type="set"> R </ci></apply> </condition>
<condition>
<apply>
<and/>
<apply><gt/><ci> x </ci><cn> 0 </cn></apply>
<apply><lt/><ci> x </ci><cn> 1 </cn></apply>
</apply>
</condition>
<apply>
<max/>
<bvar><ci> x </ci></bvar>
<condition>
<apply> <and/>
<apply><gt/><ci> x </ci><cn> 0 </cn></apply>
<apply><lt/><ci> x </ci><cn> 1 </cn></apply>
</apply>
</condition>
<apply>
<minus/>
<ci> x </ci>
<apply>
<sin/>
<ci> x </ci>
</apply>
</apply>
</apply>
0 \wedge x 
declare)The
declare construct has two primary roles. The first is to change or set the default attribute values for a specific mathematical object. The second is to establish an association between a
`name' and an object. Once a declaration is in effect, the
`name' object acquires the new attribute settings, and (if the second object is present) all the properties of the associated object.
The various attributes of the
declare element assign properties to the object being declared or determine where the declaration is in effect.
By default, the scope of a declaration is
`local' to the surrounding container element. Setting the value of the
scope attribute to
global extends the scope of the declaration to the enclosing
mathelement. As discussed in
Section 4.3.2.8 [scope], MathML contains no provision for making document-wide declarations at present, though it is anticipated that this capability will be added in future revisions of MathML, when supporting technologies become available.
declare takes one or two children. The first child, which is mandatory, is a
ci containing the identifier being declared:
<declare type="vector"> <ci> V </ci> </declare>
The second child, which is optional, is a constructor initialising the variable:
<declare type="vector">
<ci> V </ci>
<vector>
<cn> 1 </cn><cn> 2 </cn><cn> 3 </cn>
</vector>
</declare>
The constructor type and the type of the element declared must agree. For example, if the type attribute of the declaration is
fn, the second child (constructor) must be an element equivalent to an
fn element (This would include actual
fn elements,
lambda elements and any of the defined function in the basic set of content tags.) If no type is specified in the declaration then the type attribute of the declared name is set to the type of the constructor (second child) of the declaration. The type attribute of the declaration can be especially useful in the special case of the second element being a semantic tag.
All attributes are
CDATA:
type
scope
nargs
occurrence
prefix,
infix or
function-modelindications.definitionURL
encoding
The declaration
<declare type="fn" nargs="2" scope="local">
<ci> f </ci>
<apply>
<plus/>
<ci> F </ci><ci> G </ci>
</apply>
</declare>
declares f to be a two-variable function with the property that f(x, y) = (F+ G)(x, y).
The declaration
<declare type="fn">
<ci> J </ci>
<lambda>
<bvar><ci> x </ci></bvar>
<apply><ln/>
<ci> x </ci>
</apply>
</lambda>
</declare>
associates the name
J with a one-variable function defined so that
J(x) = ln
y. (Note that because of the type attribute of the
declare element, the second argument must be something of type
fn, namely a known function like
sin, an
fn construct, or a
lambdaconstruct.)
The
type attribute on the declaration is only necessary if if the type cannot be inferred from the type of the second argument.
Even when a declaration is in effect it is still possible to override attributes values selectively as in
<ci type="integer"> V
</ci>. This capability is needed in order to write statements of the form
`Let
S be a member of
S'.
Since the
declare construct is not directly rendered, most declarations are likely to be invisible to a reader. However, declarations can produce quite different effects in an application which evaluates or manipulates MathML content. While the declaration
<declare>
<ci> v </ci>
<vector>
<cn> 1 </cn>
<cn> 2 </cn>
<cn> 3 </cn>
</vector>
</declare>
is active the symbol v acquires all the properties of the vector, and even its dimension and components have meaningful values. This may affect how v is rendered by some applications, as well as how it is treated mathematically.
lambda)The lambda element is used to construct a
user-defined function from an expression and one or more free
variables. The lambda construct with n internal variables takes
n+1 children. The first n children identify the variables
that are used as placeholders in the last child for actual parameter
values. See Section 4.2.2.2 [Constructors] for further details.
The first example presents a simple lambda construct.
<lambda>
<bvar><ci> x </ci></bvar>
<apply><sin/>
<apply>
<plus/>
<ci> x </ci>
<cn> 1 </cn>
</apply>
</apply>
</lambda>
The next example constructs a one-argument function in which the argument b specifies the upper bound of a specific definite integral.
<lambda>
<bvar><ci> b </ci></bvar>
<apply>
<int/>
<bvar>
<ci> x </ci>
</bvar>
<lowlimit>
<ci> a </ci>
</lowlimit>
<uplimit>
<ci> b </ci>
</uplimit>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
</lambda>
Such constructs are often used in conjunction with
declare to construct new functions.
compose)The compose element represents the function
composition operator. Note that MathML makes no assumption about the domain
and range of the constituent functions in a composition; the domain of the
resulting composition may be empty.
To override the default semantics for the compose element, or to associate a more specific
definition for function composition, use the definitionURL and encodingattributes. See Section 4.2.3 [Functions, Operators and Qualifiers]
for further details.
<apply> <compose/> <fn><ci> f </ci></fn> <fn><ci> g </ci></fn> </apply>
<apply> <compose/> <ci type="fn"> f </ci> <ci type="fn"> g </ci> <ci type="fn"> h </ci> </apply>
<apply>
<apply><compose/>
<fn><ci> f </ci></fn>
<fn><ci> g </ci></fn>
</apply>
<ci> x </ci>
</apply>
<apply>
<fn><ci> f </ci></fn>
<apply>
<fn><ci> g </ci></fn>
<ci> x </ci>
</apply>
</apply>
ident)The ident element represents the identity
function. MathML makes no assumption about the function space in which the
identity function resides. That is, proper interpretation of the domain
(and hence range) of the identity function depends on the context in which
it is used.
To override the default semantics for the identelement, or to associate a more specific
definition, use the definitionURL and encodingattributes (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply>
<eq/>
<apply><compose/>
<fn><ci> f </ci></fn>
<apply><inverse/>
<fn><ci> f </ci></fn>
</apply>
</apply>
<ident/>
</apply>
quotient)The quotient element is the operator used for
division modulo a particular base. When the quotient operator is applied to integer arguments
a and b, the result is the `quotient of
adivided by b'. That is, quotientreturns the unique integer, q such
that a = q b + r. (In common usage,
q is called the quotient and r is the remainder.)
The quotient element takes the attribute definitionURL and encodingattributes, which can be used override the
default semantics.
The quotient element is a binary
arithmetic operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <quotient/> <ci> a </ci> <ci> b </ci> </apply>
Various mathematical applications will use this data in different ways. Editing applications might choose an image such as shown below, while a computationally based application would evaluate it to 2 when a=13 and b=5.
There is no commonly used notation for this concept. Some possible renderings are
factorial)The factorial element is used to construct factorials.
The factorial element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The factorial element is a
unary arithmetic operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <factorial/> <ci> n </ci> </apply>
If this were evaluated at n = 5 it would evaluate to 120.
divide)The divide element is the division operator.
The divide element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The divide element is a
binary arithmetic operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <divide/> <ci> a </ci> <ci> b </ci> </apply>
As a MathML expression, this does not evaluate. However, on receiving such an expression, some applications may attempt to evaluate and simplify the value. For example, when a=5 and b=2 some mathematical applications may evaluate this to 2.5 while others will treat is as a rational number.
max,
min)The elements
max and
minare used to compare the values of their arguments. They return the maximum and minimum of these values respectively.
The
max and
min elements take the
definitionURL and
encoding attributes that can be used to override the default semantics.
The
max and
min elements are
n-ary arithmetic operators (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
When the objects are to be compared explicitly they are listed as arguments to the function as in:
<apply> <max/> <ci> a </ci> <ci> b </ci> </apply>
The elements to be compared may also be described using bound variables with a
condition element and an expression to be maximised, as in:
<apply>
<min/>
<bvar><ci>x</ci></bvar>
<condition>
<apply><notin/><ci> x </ci><ci type="set"> B </ci></apply>
</condition>
<apply>
<power/>
<ci> x </ci>
<cn> 2 </cn>
</apply>
</apply>
Note that the bound variable must be stated even if it might be implicit in conventional notation. In MathML1.0, the bound variable and expression to be evaluated (x) could be omitted in the example below: this usage is deprecated in MathML2.0 in favour of explicitly stating the bound variable and expression in all cases:
<apply>
<bvar><ci>x</ci></bvar>
<max/>
<condition>
<apply><and/>
<apply><in/><ci>x</ci><ci type="set">B</ci></apply>
<apply><notin/><ci>x</ci><ci type="set">C</ci></apply>
</apply>
</condition>
<ci>x</ci>
</apply>
minus)The minus element is the subtraction operator.
The minus element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The minus element can be used as a unary
arithmetic operator (e.g. to represent - x), or as a
binary arithmetic operator (e.g. to represent x-
y).
<apply> <minus/> <ci> x </ci> <ci> y </ci> </apply>
If this were evaluated at x=5 and y=2 it would yield 3.
plus)The plus element is the addition operator.
The plus element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The plus element is an n-ary arithemtic
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <plus/> <ci> x </ci> <ci> y </ci> <ci> z </ci> </apply>
If this were evaluated at x = 5, y = 2 and z = 1 it would yield 8.
power)The power element is a generic exponentiation
operator. That is, when applied to arguments a and b, it
returns the value the
`
a to the power of
b'.
The
power element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
power element is an
binary arithmetic operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <power/> <ci> x </ci> <cn> 3 </cn> </apply>
If this were evaluated at x= 5 it would yield 125.
rem)The rem element is the operator that returns the
`remainder' of a division modulo a particular base. When the
rem operator is applied to integer arguments
a and b, the result is the `remainder of
adivided by b'. That is, rem returns the unique integer, r such that
a = q b+ r, where r <
q. (In common usage, q is called the quotient and
r is the remainder.)
The
rem element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
rem element is a
binary arithmetic operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <rem/> <ci> a </ci> <ci> b </ci> </apply>
If this were evaluated at a = 15 and b = 8 it would yield 7.
times)The
times element is the multiplication operator.
times takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
<apply> <times/> <ci> a </ci> <ci> b </ci> </apply>
If this were evaluated at a = 5.5 and b = 3 it would yield 16.5.
root)The root element is used to construct roots. The
kind of root to be taken is specified by a degreeelement, which should be given as the first child
of the apply element enclosing the root element. Thus, square roots correspond to the case
where degree contains the value 2, cube roots
correspond to 3, and so on. If no degree is
present, a default value of 2 is used.
The root element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The root element is an
operator taking qualifiers (see
Section 4.2.3.2 [Operators taking Qualifiers]).
The nth root of a is is given by
<apply> <root/> <degree><ci type='integer'> n </ci></degree> <ci> a </ci> </apply>
![\sqrt[n]{a}](http://www.w3.org/TR/2000/WD-MathML2-20000328/image/f4037.gif)
gcd)The gcd element is used to denote the greatest
common divisor of its arguments.
The gcd takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The gcd element is an
n-ary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <gcd/> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply>
If this were evaluated at a = 15, b = 21, c = 48 it would yield 3.
and)The and element is the boolean
`and' operator.
The
and element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
and element is an
n-ary logical operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <and/> <ci> a </ci> <ci> b </ci> </apply>
If this were evaluated and both
a and
b had truth values of
true, then the value would be
true.
or)The or element is the boolean
`or' operator.
The
or element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
or element is an
n-ary logical operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <or/> <ci> a </ci> <ci> b </ci> </apply>
xor)The xor element is the boolean `exclusive
or'operator.
xor takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The xor element is an n-ary logical
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <xor/> <ci> a </ci> <ci> b </ci> </apply>
not)The not operator is the boolean
`not' operator.
The
not element takes the attribute
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
not element is a
unary logical operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <not/> <ci> a </ci> </apply>
implies)The implies element is the boolean relational operator
`implies'.
The
implies element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
implies element is a
binary logical operator (see
Section 4.2.4 [Relations]).
<apply> <implies/> <ci> A </ci> <ci> B </ci> </apply>
Mathematical applications designed for the evaluation of such expressions would evaluate this to
true when
a =
false and
b =
true.
forall)The
forall element represents the universal quantifier of logic. It must used in conjunction with one or more bound variables, an optional
condition element, and an assertion, which may either take the form of an
apply or
reln element.
The
forall element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
forall element is an
quantifier (see
Section 4.2.3.2 [Operators taking Qualifiers]).
The first example encodes a simple identity.
<apply>
<forall/>
<bvar><ci> x </ci></bvar>
<apply><eq/>
<apply>
<minus/><ci> x </ci><ci> x </ci>
</apply>
<cn>0</cn>
</apply>
</apply>
The next example is more involved, and makes use of an optional
condition element.
<apply>
<forall/>
<bvar><ci> p </ci></bvar>
<bvar><ci> q </ci></bvar>
<condition>
<apply><and/>
<apply><in/><ci> p </ci><ci type="set"> Q </ci></apply>
<apply><in/><ci> q </ci><ci type="set"> Q </ci></apply>
<apply><lt/><ci> p </ci><ci> q </ci></apply>
</apply>
</condition>
<apply><lt/>
<ci> p </ci>
<apply>
<power/>
<ci> q </ci>
<cn> 2 </cn>
</apply>
</apply>
</apply>
The final example uses both the
forall and
exists quantifiers.
<apply>
<forall/>
<bvar><ci> n </ci></bvar>
<condition>
<apply><and/>
<apply><gt/><ci> n </ci><cn> 0 </cn></apply>
<apply><in/><ci> n </ci><ci type="set"> Z </ci></apply>
</apply>
</condition>
<apply>
<exists/>
<bvar><ci> x </ci></bvar>
<bvar><ci> y </ci></bvar>
<bvar><ci> z </ci></bvar>
<condition>
<apply><and/>
<apply><in/><ci> x </ci><ci type="set"> Z </ci></apply>
<apply><in/><ci> y </ci><ci type="set"> Z </ci></apply>
<apply><in/><ci> z </ci><ci type="set"> Z </ci></apply>
</apply>
</condition>
<apply>
<eq/>
<apply>
<plus/>
<apply><power/><ci> x </ci><ci> n </ci></apply>
<apply><power/><ci> y </ci><ci> n </ci></apply>
</apply>
<apply><power/><ci> z </ci><ci> n </ci></apply>
</apply>
</apply>
</apply>
0, n \in \mathbb{Z}: \exists x \in \mathbb{Z}, y \in \mathbb{Z}, z \in \mathbb{Z}: x^n+y^n=z^n" align="middle">exists)The exists element represents the existential
quantifier of logic. It must used in conjuction with one or more bound
variables, an optional condition element, and an
assertion, which may either take the form of an apply or reln element.
The exists element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The exists element is an
quantifier (see Section 4.2.3.2 [Operators taking Qualifiers]).
The following example encodes the sense of the expression `there exists an x such that f(x) = 0'.
<apply>
<exists/>
<bvar><ci> x </ci></bvar>
<apply><eq/>
<apply>
<fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
<cn>0</cn>
</apply>
</apply>
abs)The abs element represents the absolute value of
a real quantity or the modulus of a complex quantity.
The abs element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The abs element is a unary arithmetic
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
The following example encodes the absolute value of x.
<apply> <abs/> <ci> x </ci> </apply>
conjugate)The conjugate element represents the complex
conjugate of a complex quantity.
The conjugate element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The conjugate element is a unary
arithmetic operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
The following example encodes the conjugate of x + i y.
<apply>
<conjugate/>
<apply>
<plus/>
<ci> x </ci>
<apply><times/>
<cn> ⅈ </cn>
<ci> y </ci>
</apply>
</apply>
</apply>
arg)The arg operator (introduced in MathML 2.0))
gives the `argument' of a complex number, which is the angle
(in radians) it makes with the positive real axis. Real negative numbers
have argument equal to + .
The arg element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The arg element is a unary arithmetic
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
The following example encodes the argument operation on x + i y.
<apply>
<arg/>
<apply><plus/>
<ci> x </ci>
<apply><times/>
<cn> ⅈ </cn>
<ci> y </ci>
</apply>
</apply>
</apply>
real)The real operator (introduced in MathML 2.0)
gives the real part of a complex number, that is the x component in
x + i y
The real element takes the attributes encoding, definitionURL that can be used to override the
default semantics.
The real element is a unary arithmetic
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
The following example encodes the real operation on x + i y.
<apply>
<real/>
<apply><plus/>
<ci> x </ci>
<apply><times/>
<cn> ⅈ </cn>
<ci> y </ci>
</apply>
</apply>
</apply>
A MathML-aware evaluation system would return the x component, suitably encoded.
imaginary)The imaginary operator (introduced in MathML
2.0) gives the imaginary part of a complex number, that is the y component
in x + i y
The imaginary element takes the attributes encoding, definitionURL that can be used to override the
default semantics.
The imaginary element is a unary
arithmetic operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
The following example encodes the imaginary operation on x + i y.
<apply>
<imaginary/>
<apply><plus/>
<ci> x </ci>
<apply><times/>
<cn> ⅈ </cn>
<ci> y </ci>
</apply>
</apply>
</apply>
A MathML-aware evaluation system would return the y component, suitably encoded.
lcm)The
lcm element (introduced in MathML 2.0) is used to denote the lowest common
multiple of its arguments.
The
lcm takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
lcm element is an
n-ary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <lcm/> <ci> a </ci> <ci> b </ci> <ci> c </ci> </apply>
If this were evaluated at a = 2, b = 4, c = 6 it would yield 12.
eq)The eq element is the relational operator
`equals'.
The
eq element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
equals element is an
n-ary relation (see
Section 4.2.3.2 [Operators taking Qualifiers]).
<apply> <eq/> <ci> a </ci> <ci> b </ci> </apply>
If this were tested at a = 5.5 and b = 6 it would
yield the truth value false.
neq)The neq element is the `not equal
to' relational operator.
neq takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The neq element is an binary
relation (see Section 4.2.4 [Relations]).
<apply> <neq/> <ci> a </ci> <ci> b </ci> </apply>
If this were tested at a = 5.5 and b = 6 it would
yield the truth value true.
gt)The gt element is the `greater
than' relational operator.
The gt element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The gt element is an n-ary
relation (see Section 4.2.4 [Relations]).
<apply> <gt/> <ci> a </ci> <ci> b </ci> </apply>
If this were tested at
a = 5.5 and
b = 6 it would yield the truth value
false.
b" align="middle">
lt)The lt element is the `less than'
relational operator.
The lt element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The lt element is an n-ary
relation (see Section 4.2.4 [Relations]).
<apply> <lt/> <ci> a </ci> <ci> b </ci> </apply>
If this were tested at a = 5.5 and b = 6 it would yield the truth value `true'.
geq)The geq element is the relational operator
`greater than or equal'.
The
geq element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
geq element is an
n-ary relation (see
Section 4.2.4 [Relations]).
<apply> <geq/> <ci> a </ci> <ci> b </ci> </apply>
If this were tested for
a = 5.5 and
b = 5.5 it would yield the truth value
true.
leq)The leq element is the relational operator
`less than or equal'.
The
leq element takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
The
leq element is an
n-ary relation (see
Section 4.2.4 [Relations]).
<apply> <leq/> <ci> a </ci> <ci> b </ci> </apply>
If
a = 5.4 and
b = 5.5 this will yield the truth value
true.
equivalent)The equivalent element is the
`equivalence' relational operator.
The
equivalent element takes the attributes
encoding, definitionURL that can be used to override the default semantics.
The
equivalent element is an
n-ary relation (see
Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<equivalent/>
<ci> a </ci>
<apply>
<not/>
<apply> <not/> <ci> a </ci> </apply>
</apply>
</apply>
This yields the truth value
true for all values of
a.
approx)The approx element is the relational operator
`approximately equal'.
The
approx element takes the attributes
encoding, definitionURL that can be used to override the default semantics.
The
approx element is a
binary relation (see
Section 4.2.3.2 [Operators taking Qualifiers]).
<apply> <approx/> <cn type="rational"> 22 <sep/> 7 </cn> <cn type="constant"> π </cn> </apply>
int)The int element is the operator element for an
integral. The lower limit, upper limit and bound variable are given by
(optional) child elements, lowlimit, uplimit and bvar in the
enclosing apply element. The integrand is also
specified as a child element of the enclosing apply
element.
The domain of integration may alternatively be specified by using an interval element, or by a condition element. In such cases, if a bound variable
of integration is intended, it must be specified explicitly. (The
condition may involve more than one symbol.)
The int element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The int element is an operator taking
qualifiers (see Section 4.2.3.2 [Operators taking Qualifiers]).
This example specifies a lowlimit, uplimit, and bvar.
<apply>
<int/>
<bvar>
<ci> x </ci>
</bvar>
<lowlimit>
<cn> 0 </cn>
</lowlimit>
<uplimit>
<ci> a </ci>
</uplimit>
<apply>
<fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
This example specifies the domain of integration with an
interval element.
<apply>
<int/>
<bvar>
<ci> x </ci>
</bvar>
<interval>
<ci> a </ci>
<ci> b </ci>
</interval>
<apply><cos/>
<ci> x </ci>
</apply>
</apply>
The final example specifies the domain of integration with an
condition element.
<apply>
<int/>
<bvar>
<ci> x </ci>
</bvar>
<condition>
<apply><in/>
<ci> x </ci>
<ci type="set"> D </ci>
</apply>
</condition>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
diff)The diff element is the differentiation operator
element for functions of a single real variable. The bound variable is
given by a bvar element that is a child of the
containing apply element. The bvar elements may also contain a degree element, which specifies the order of the
derivative to be taken.
The diff element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The diff element is an operator taking
qualifiers (see Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<diff/>
<bvar>
<ci> x </ci>
</bvar>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
partialdiff)The partialdiff element is the partial
differentiation operator element for functions of several real
variables. The bound variables are given by bvar
elements, which are children of the containing apply element. The bvarelements
may also contain a degree element, which specifies
the order of the partial derivative to be taken in that variable.
The partialdiff element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The partialdiff element is an operator
taking qualifiers (see Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<partialdiff/>
<bvar>
<ci> x </ci>
<degree>
<cn> 2 </cn>
</degree>
</bvar>
<bvar>
<ci> y </ci>
</bvar>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
<ci> y </ci>
</apply>
</apply>
lowlimit)The lowlimit element is the container element
used to indicate the `lower limit' of an operator using
qualifiers. For example, in an integral, it can be used to specify the
lower limit of integration. Similarly, it is also used to specify the lower
limit of an index for sums and products.
The meaning of the lowlimit element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 [Operators taking Qualifiers].
<apply>
<int/>
<bvar>
<ci> x </ci>
</bvar>
<lowlimit>
<ci> a </ci>
</lowlimit>
<uplimit>
<ci> b </ci>
</uplimit>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
The default rendering of the
lowlimit element and its contents depends on the context. In the preceding example, it should be rendered as a subscript to the integral sign:
Consult the descriptions of individual operators that make use of the
lowlimit construct for default renderings.
uplimit)The uplimit element is the container element
used to indicate the `upper limit' of an operator using
qualifiers. For example, in an integral, it can be used to specify the
upper limit of integration. Similarly, it is also used to specify the upper
limit of an index for sums and products.
The meaning of the uplimit element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 [Operators taking Qualifiers].
<apply>
<int/>
<bvar>
<ci> x </ci>
</bvar>
<lowlimit>
<ci> a </ci>
</lowlimit>
<uplimit>
<ci> b </ci>
</uplimit>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
The default rendering of the uplimit element and
its contents depends on the context. In the preceding example, it should be
rendered as a superscript to the integral sign:
Consult the descriptions of individual operators that make use of the
uplimit construct for default renderings.
bvar)The bvar element is the container element for
the `bound variable' of an operation. For example, in an
integral it specifies the variable of integration. In a derivative, it
indicates which variable with respect to which a function is being
differentiated. When the bvar element is used to
quantify a derivative, the bvar element may contain
a child degree element that specifies the order of
the derivative with respect to that variable. The bvarelement is also used for the internal variable in
sums and products and for the bound variable used with the universal and
existential quantifiers forall and exists.
The meaning of the bvar element depends on the
context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 [Operators taking Qualifiers].
<apply>
<diff/>
<bvar>
<ci> x </ci>
<degree>
<cn> 2 </cn>
</degree>
</bvar>
<apply>
<power/>
<ci> x </ci>
<cn> 4 </cn>
</apply>
</apply>
<apply>
<int/>
<bvar><ci> x </ci></bvar>
<condition>
<apply><in/><ci> x </ci><ci> D </ci></apply>
</condition>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
The default rendering of the
bvar element and its contents depends on the context. In the preceding examples, it should be rendered as the
x in the d
x of the integral, and as the
x in the denominator of the derivative symbol:
Note that in the case of the derivative, the default rendering of the
degree child of the
bvar element is as an exponent.
Consult the descriptions of individual operators that make use of the
bvar construct for default renderings.
degree)The degree element is the container element for
the `degree' or `order' of an operation. There
are a number basic mathematical constructs that come in families, such as
derivatives and moments. Rather than introduce special elements for each of
these families, MathML uses a single general construct, the degree element for this concept of
`order'.
The meaning of the
degree element depends on the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking qualifiers, consult
Section 4.2.3.2 [Operators taking Qualifiers].
<apply>
<partialdiff/>
<bvar>
<ci> x </ci>
<degree>
<ci> n </ci>
</degree>
</bvar>
<bvar>
<ci> y </ci>
<degree>
<ci> m </ci>
</degree>
</bvar>
<apply><sin/>
<apply> <times/>
<ci> x </ci>
<ci> y </ci>
</apply>
</apply>
</apply>
The default rendering of the
degree element and its contents depends on the context. In the preceding example, the
degree elements would be rendered as the exponents in the differentiation symbols:
Consult the descriptions of individual operators that make use of the
degree construct for default renderings.
divergence)The divergence element is the vector calculus
divergence operator, often called div.
The divergence element takes the attributes encoding, definitionURL that can be used to override the
default semantics.
The divergence element is an
unary calculus operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <divergence/> <ci> a </ci> </apply>
If a is a vector field defined inside a closed surface S enclosing a volume V, then the divergence of a is given by
<apply>
<limit/>
<bvar>
<ci> V </ci>
</bvar>
<condition>
<apply>
<tendsto/>
<ci> V </ci>
<cn> 0 </cn>
</apply>
</condition>
<apply>
<divide/>
<apply><int encoding="text" definitionURL="SurfaceIntegrals.htm"/>
<bvar>
<ci> S</ci>
</bvar>
<ci> a </ci>
</apply>
<ci> V </ci>
</apply>
</apply>
grad)The grad element is the vector calculus gradient
operator, often called grad.
The grad element takes the attributes encoding, definitionURL that can be used to override the
default semantics.
The grad element is an unary calculus
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <grad/> <ci> f</ci> </apply>
Where for example f is a scalar function and f(x, y, z) = k defines a surface S
curl)The curl element is the vector calculus curl operator.
The curl element takes the attributes encoding, definitionURL that can be used to override the
default semantics.
The curl element is an unary calculus
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <curl/> <ci> a </ci> </apply>
Where for example a is a vector.
laplacian)The laplacian element is the vector calculus
laplacian operator.
The laplacian element takes the attributes encoding, definitionURL that can be used to override the
default semantics.
The laplacian element is an unary calculus
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply>
<eq/>
<apply><laplacian/>
<ci> f </ci>
</apply>
<apply>
<divergence/>
<apply><grad/>
<ci> f </ci>
</apply>
</apply>
</apply>
Where for example f is a vector
set)The
set element is the container element that constructs a set of elements. The elements of a set can be defined either by explicitly listing the elements, or by using the
bvar and
condition elements.
The
set element is a
constructor element (see
Section 4.2.2.2 [Constructors]).
<set> <ci> b </ci> <ci> a </ci> <ci> c </ci> </set>
<set>
<bvar><ci> x </ci></bvar>
<condition>
<apply><lt/>
<ci> x </ci>
<cn> 5 </cn>
</apply>
</condition>
</set>
list)The
list element is the container element that constructs a list of elements. Elements can be defined either by explicitly listing the elements, or by using the
bvar and
condition elements.
Lists differ from sets in that there is an explicit order to the elements. Two orders are supported: lexicographic and numeric. The kind of ordering that should be used is specified by the
order attribute.
The
list element is a
constructor element (see
Section 4.2.2.2 [Constructors]).
<list> <ci> a </ci> <ci> b </ci> <ci> c </ci> </list>
<list order="numeric">
<bvar><ci> x </ci></bvar>
<condition>
<apply><lt/>
<ci> x </ci>
<cn> 5 </cn>
</apply>
</condition>
</list>
![[ a, b, c ]](http://www.w3.org/TR/2000/WD-MathML2-20000328/image/f4076.gif)
![[ x \mid x < 5 ]](http://www.w3.org/TR/2000/WD-MathML2-20000328/image/f4077.gif)
union)The union element is the operator for a
set-theoretic union or join of two (or more) sets.
The union attribute takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The union element is an n-ary set
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <union/> <ci> A </ci> <ci> B </ci> </apply>
intersect)The intersect element is the operator for the
set-theoretic intersection or meet of two (or more) sets.
The intersect element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The intersect element is an n-ary set
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <intersect/> <ci type="set"> A </ci> <ci type="set"> B </ci> </apply>
in)The in element is the relational operator used
for a set-theoretic inclusion (`is in' or `is a member
of').
The in element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The in element is a binary set
relation (see Section 4.2.4 [Relations]).
<apply> <in/> <ci> a </ci> <ci type="set"> A </ci> </apply>
notin)The notin element is the relational operator
element used for set-theoretic exclusion (`is not in' or
`is not a member of').
The notin element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The notin element is a binary set
relation (see Section 4.2.4 [Relations]).
<apply> <notin/> <ci> a </ci> <ci> A </ci> </apply>
subset)The subset element is the relational operator
element for a set-theoretic containment (`is a subset
of').
The subset element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The subset element is an n-ary set
relation (see Section 4.2.4 [Relations]).
<apply> <subset/> <ci> A </ci> <ci> B </ci> </apply>
prsubset)The prsubset element is the relational operator
element for set-theoretic proper containment (`is a proper subset
of').
The prsubset element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The subset element is an n-ary set
relation (see Section 4.2.4 [Relations]).
<apply> <prsubset/> <ci> A </ci> <ci> B </ci> </apply>
notsubset)The notsubset element is the relational operator
element for the set-theoretic relation `is not a subset
of'.
The notsubset element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The notsubset element is a binary set
relation (see Section 4.2.4 [Relations]).
<apply> <notsubset/> <ci> A </ci> <ci> B </ci> </apply>
notprsubset)The notprsubset element is the operator element
for the set-theoretic relation `is not a proper subset
of'.
The notprsubset takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The notprsubset element is a binary set
relation (see Section 4.2.4 [Relations]).
<apply> <notprsubset/> <ci> A </ci> <ci> B </ci> </apply>
setdiff)The setdiff element is the operator element for
a set-theoretic difference of two sets.
The setdiff element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The setdiff element is a binary set
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <setdiff/> <ci> A </ci> <ci> B </ci> </apply>
card)The card element is the operator element for
deriving the size or cardinality of a set
The card element takes the attributes definitionURL, encoding that can be used to override the
default semantics.
The card element is a unary set
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply>
<eq/>
<apply><card/>
<ci> A </ci>
</apply>
<ci> 5 </ci>
</apply>
where A is a set with 5 elements.
sum)The sum element denotes the summation
operator. Upper and lower limits for the sum, and more generally a domains
for the bound variables are specified using uplimit, lowlimit or a condition on the bound variables. The index for the
summation is specified by a bvar element.
The sum element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The sum element is an operator taking
qualifiers (see Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<sum/>
<bvar>
<ci> x </ci>
</bvar>
<lowlimit>
<ci> a </ci>
</lowlimit>
<uplimit>
<ci> b </ci>
</uplimit>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
<apply>
<sum/>
<bvar>
<ci> x </ci>
</bvar>
<condition>
<apply> <in/>
<ci> x </ci>
<ci type="set"> B </ci>
</apply>
</condition>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
product)The product element denotes the product
operator. Upper and lower limits for the product, and more generally a
domains for the bound variables are specified using uplimit, lowlimit or a condition on the bound variables. The index for the
product is specified by a bvar element.
The product element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The product element is an operator taking
qualifiers (see Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<product/>
<bvar>
<ci> x </ci>
</bvar>
<lowlimit>
<ci> a </ci>
</lowlimit>
<uplimit>
<ci> b </ci>
</uplimit>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
<apply>
<product/>
<bvar>
<ci> x </ci>
</bvar>
<condition>
<apply> <in/>
<ci> x </ci>
<ci type="set"> B </ci>
</apply>
</condition>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
</apply>
</apply>
limit)The limit element represents the operation of
taking a limit of a sequence. The limit point is expressed by specifying a
lowlimit and a bvar, or by
specifying a condition on one or more bound
variables.
The limit element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The limit element is an operator taking
qualifiers (see Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<limit/>
<bvar>
<ci> x </ci>
</bvar>
<lowlimit>
<cn> 0 </cn>
</lowlimit>
<apply><sin/>
<ci> x </ci>
</apply>
</apply>
<apply>
<limit/>
<bvar>
<ci> x </ci>
</bvar>
<condition>
<apply>
<tendsto type="above"/>
<ci> x </ci>
<ci> a </ci>
</apply>
</condition>
<apply><sin/>
<ci> x </ci>
</apply>
</apply>
tendsto)The tendsto element is used to express the
relation that a quantity is tending to a specified value.
The tendsto element takes the attributes type to set the direction from which the the limiting
value is approached and the definitionURL and encoding attributes, which can be used to override the
default semantics.
The tendsto element is a binary relational
operator (see Section 4.2.4 [Relations]).
<apply>
<tendsto type="above"/>
<apply>
<power/>
<ci> x </ci>
<cn> 2 </cn>
</apply>
<apply>
<power/>
<ci> a </ci>
<cn> 2 </cn>
</apply>
</apply>
To express (x,
y)
(f(x,
y),
g(x,
y)), one might use vectors, as in:
<apply>
<tendsto/>
<vector>
<ci> x </ci>
<ci> y </ci>
</vector>
<vector>
<apply><fn><ci> f </ci></fn>
<ci> x </ci>
<ci> y </ci>
</apply>
<apply><fn><ci> g </ci></fn>
<ci> x </ci>
<ci> y </ci>
</apply>
</vector>
</apply>
The names of the common trigonometric functions supported by MathML are listed below. Since their standard interpretations are widely known, they are discussed as a group.
sin |
cos |
tan |
sec |
csc |
cot |
sinh |
cosh |
tanh |
sech |
csch |
coth |
arcsin |
arccos |
arctan |
arccosh |
arccot |
arccoth |
arccsc |
arccsch |
arcsec |
arcsech |
arcsinh |
arctanh |
These operator elements denote the standard trigonometrical functions.
These elements all take the definitionURL and encoding attributes, which can be used to override the
default semantics.
They are all unary trigonometric operators. (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <sin/> <ci> x </ci> </apply>
<apply>
<sin/>
<apply>
<plus/>
<apply><cos/>
<ci> x </ci>
</apply>
<apply>
<power/>
<ci> x </ci>
<cn> 3 </cn>
</apply>
</apply>
</apply>
exp)The exp element represents the exponential
function associated with the inverse of the ln
function. In particular, exp(1) is approximately 2.718281828.
The exp element takes the definitionURL and encodingattributes, which may be used to override the
default semantics.
The exp element is a unary arithmetic
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <exp/> <ci> x </ci> </apply>
ln)The ln element is the natural logarithm
operator.
The ln element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The ln element is an unary calculus
operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <ln/> <ci> a </ci> </apply>
If a = e this will yield the value 1.
log)The log element is the operator that returns a
logarithm to a given base. The base may be specified using a logbase element, which should be the first element
following log, i.e. the second child of the
containing apply element. If the logbase element is not present, a default base of 10 is
assumed.
The log element takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
The log element can be used as either an
operator taking qualifiers or a unary calculus
operator (see Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<log/>
<logbase>
<cn> 3 </cn>
</logbase>
<ci> x </ci>
</apply>
This markup represents `the base 3 logarithm of x'. For
natural logarithms base e, the ln element should be
used instead.
mean)
mean is the operator element for a mean or average.
mean takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
mean is an n-ary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <mean/> <ci> X </ci> </apply>
or
sdev)
sdev is the operator element for the standard deviation.
sdev takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
sdev is an n-ary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <sdev/> <ci> X </ci> </apply>
variance)
variance is the operator element for the
statistical variance.
variance takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
variance is an n-ary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <variance/> <ci> X </ci> </apply>
median)
median is the operator element for the median.
median takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
median is an n-ary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <median/> <ci> X </ci> </apply>
mode)
mode is the operator for the statistical mode.
mode takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
mode is an n-ary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <mode/> <ci> X </ci> </apply>
moment)The moment element represents statistical
moments. Use degree for the n in
`
n-th moment'.
moment takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
moment is an operator taking qualifiers (see
Section 4.2.3.2 [Operators taking Qualifiers]).
<apply>
<moment/>
<degree>
<cn> 3 </cn>
</degree>
<ci> X </ci>
</apply>
vector)
vector is the container element for a
vector. The child elements form the components of the vector.
For purposes of interaction with matrices and matrix multiplication, vectors are regarded as equivalent to a matrix consisting of a single column, and the transpose of a vector behaves the same as a matrix consisting of a single row.
vector is a constructor element (see
Section 4.2.2.2 [Constructors]).
<vector> <cn> 1 </cn> <cn> 2 </cn> <cn> 3 </cn> <ci> x </ci> </vector>
(1, 2, 3, x)
matrix)The matrix element is the container element for
matrix rows, which are represented by matrixrow. The matrixrows
contain the elements of a matrix.
matrix is a constructor element (see
Section 4.2.2.2 [Constructors]).
<matrix>
<matrixrow>
<cn> 0 </cn> <cn> 1 </cn> <cn> 0 </cn>
</matrixrow>
<matrixrow>
<cn> 0 </cn> <cn> 0 </cn> <cn> 1 </cn>
</matrixrow>
<matrixrow>
<cn> 1 </cn> <cn> 0 </cn> <cn> 0 </cn>
</matrixrow>
</matrix>
matrixrow)The matrixrow element is the container element
for the rows of a matrix.
matrixrow is a constructor element (see
Section 4.2.2.2 [Constructors]).
<matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow> <matrixrow> <cn> 3 </cn> <ci> x </ci> </matrixrow>
Matrix rows are not directly rendered by themselves outside of the context of a matrix.
determinant)The determinant element is the operator for constructing the determinant of a matrix.
determinant takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
determinant is a unary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <determinant/> <ci type="matrix"> A </ci> </apply>
transpose)The transpose element is the operator for
constructing the transpose of a matrix.
transpose takes the definitionURL and encodingattributes, which can be used to override the
default semantics.
transpose is a
unary operator (see
Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <transpose/> <ci type="matrix"> A </ci> </apply>
selector)The selector element is the operator for
indexing into vectors matrices and lists. It accepts one or more
arguments. The first argument identifies the vector, matrix or list from
which the selection is taking place, and the second and subsequent
arguments, if any, indicate the kind of selection taking place.
When selector is used with a single argument, it
should be interpreted as giving the sequence of all elements in the list,
vector or matrix given. The ordering of elements in the sequence for a
matrix is understood to be first by column, then by row. That is, for a
matrix ( ai,j), where the indices
denote row and column, the ordering would be a 1,1,
a 1,2, ... a 2,1, a
2,2 ... etcetera.
When three arguments are given, the last one is ignored for a list or vector, and in the case of a matrix, the second and third arguments specify the row and column of the selected element.
When two arguments are given, and the first is a vector or list, the second argument specifies an element in the list or vector. When a matrix and only one index i is specified as in
<apply>
<selector/>
<matrix>
<matrixrow>
<cn> 1 </cn> <cn> 2 </cn>
</matrixrow>
<matrixrow>
<cn> 3 </cn> <cn> 4 </cn>
</matrixrow>
</matrix>
<cn> 1 </cn>
</apply>
it refers to the i-th matrixrow. Thus, the preceding example selects the following row:
<matrixrow> <cn> 1 </cn> <cn> 2 </cn> </matrixrow>
selector takes the
definitionURL and
encodingattributes, which can be used to override the default semantics.
selector is classified as an n-ary linear algebra operator even though it can take only one, two, or three arguments.
<apply> <selector/> <ci type="matrix"> A </ci> <cn> 3 </cn> <cn> 2 </cn> </apply>
The selector construct renders the same as the
expression it selects.
vectorproduct)The vectorproduct is the operator element for
deriving the vector product of two vectors
The vectorproduct element takes the attributes
definitionURL, encoding that can be used to override
the default semantics.
The vectorproduct element is a binary
vector operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply>
<eq/>
<apply><vectorproduct/>
<ci type="vector"> A </ci>
<ci type="vector"> B </ci>
</apply>
<apply><times/>
<ci> a </ci>
<ci> b </ci>
<apply><sin/>
<ci> θ </ci>
</apply>
</apply>
</apply>
where A and B are vectors, a, b are the magnitudes of
A, B and 
is
the angle between A and B.
scalarproduct)The scalarproduct is the operator element for
deriving the scalar product of two vectors
The scalarproduct element takes the attributes
definitionURL, encoding that can be used to override
the default semantics.
The scalarproduct element is a binary
vector operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply>
<eq/>
<apply><scalarproduct/>
<ci type="vector"> A </ci>
<ci type="vector">B </ci>
</apply>
<apply><times/>
<ci> a </ci>
<ci> b </ci>
<apply><cos/>
<ci> θ </ci>
</apply>
</apply>
</apply>
where A and B are vectors, a, b are the magnitudes of
A, B and is
the angle between A and B.
outerproduct)The outerproduct is the operator element for
deriving the outer product of two vectors
The outerproduct element takes the attributes
definitionURL, encoding that can be used to override
the default semantics.
The outerproduct element is a
binary vector operator (see Section 4.2.3 [Functions, Operators and Qualifiers]).
<apply> <outerproduct/> <ci type="vector">A</ci> <ci type="vector">B</ci> </apply>
where A and B are vectors.
This section explains the use of the semantic mapping elements semantics, annotation and annotation-xml.
annotation)The annotation element is the container element
for a semantic annotation in a non-XML format.
The annotation element takes the attribute encoding to define the encoding being used.
The annotation element is a semantic mapping
element. It is always used with semantics.
<semantics>
<apply>
<plus/>
<apply><sin/>
<ci> x </ci>
</apply>
<cn> 5 </cn>
</apply>
<annotation encoding="TeX">
\sin x + 5
</annotation>
</semantics>
None. The information contained in annotations may optionally be used by a renderer able to process the kind of annotation given.
semantics)The semantics element is the container element
that associates additional representations with a given MathML
construct. The semantics element has as its first
child the expression being annotated, and the subsequent children are the
annotations. There is no restriction on the kind of annotation that can be
attached using the semantics element. For example, one might give a TEX
encoding, or computer algebra input in an annotation.
The representations that are XML based are enclosed in an annotation-xml element while those representations that
are to be parsed as PCDATA are enclosed in an annotation element.
The semantics element takes the definitionURL and encodingattributes, which can be used to reference an
external source for some or all of the semantic information.
An important purpose of the semantics construct
is to associate specific semantics with a particular presentation, or
additional presentation information with a content construct. The default
rendering of a semantics element is the default
rendering of its first child. When a MathML-presentation annotation is
provided, a MathML renderer may optionally use this information to render
the MathML construct. This would typically be the case when the first child
is a MathML content construct and the annotation is provided to give a
preferred rendering differing from the default for the content
elements.
Use of semantics to attach additional
information in-line to a MathML construct can be contrasted with use of the
csymbol for referencing external semantics. See
Section 4.4.1.3 [Externally defined symbol (csymbol)]
The semantics element is a semantic mapping element.
<semantics>
<apply>
<plus/>
<apply>
<sin/>
<ci> x </ci>
</apply>
<cn> 5 </cn>
</apply>
<annotation encoding="Maple">
sin(x) + 5
</annotation>
<annotation-xml encoding="MathML-Presentation">
...
...
</annotation-xml>
<annotation encoding="Mathematica">
Sin[x] + 5
</annotation>
<annotation encoding="TeX">
\sin x + 5
</annotation>
<annotation-xml encoding="OpenMath">
<OMA>...</OMA>
</annotation-xml>
</semantics>
The default rendering of a semantics element is
the default rendering of its first child.
annotation-xml)The annotation-xml container element is used to
contain representations that are XML based. It is always used together with
the semantics element, and takes the attribute encoding to define the encoding being used.
annotation-xml is a semantic mapping element.
<semantics>
<apply>
<plus/>
<apply><sin/>
<ci> x </ci>
</apply>
<cn> 5 </cn>
</apply>
<annotation-xml encoding="OpenMath">
<OMA><OMS name="plus" cd="arith1"/>
<OMA><OMS name="sin" cd="transc1"/>
<OMV name="x"/>
</OMA>
<OMI>5</OMI>
</OMA>
</annotation-xml>
</semantics>
See also the discussion of semantics above.
None. The information may optionally be used by a renderer able to process the kind of annotation given.
This section explains the use of the Constant and Symbol elements.
integers)
integers represents the set of all integers.
<apply> <in/> <cn type="integer"> 42 </cn> <integers/> </apply>
reals)
reals represents the set of all real numbers.
<apply> <in/> <cn type="real"> 44.997 </cn> <reals/> </apply>
reals)
rationals represents the set of all rational numbers.
<apply> <in/> <cn type="rational"> 22 <sep/>7</cn> <rationals/> </apply>
naturalnumbers)
naturalnumbers represents the set of all natural
numbers, ie. non-negative integers.
<apply> <in/> <cn type="integer">1729</cn> <naturalnumbers/> </apply>
complexes)
complexes represents the set of all complex
numbers, ie. numbers which may have a real and an imaginary part.
complexes represents the set of all complex numbers, ie. numbers which may have a real and an imaginary part.
<apply> <in/> <ci type="complex">17<sep/>29</ci> <complexes/> </apply>
primes)
primes represents the set of all natural prime
numbers, ie. integers greater than 1 which have no positive integer factor
other than themselves and 1.
<apply> <in/> <cn type="integer">17</cn> <primes/> </apply>
exponentiale)
exponentiale represents the mathematical
constant which is the exponential base of the natural logarithms, commonly
written e . It is approximately 2.718281828..
<apply> <eq/>
<apply>
<ln/>
<exponentiale/>
</apply>
<cn>1</cn>
</apply>
imaginaryi)
imaginaryi represents the mathematical constant
which is the square root of -1, commonly written i.
<apply> <eq/>
<apply>
<power/>
<imaginaryi/>
<cn>2</cn>
</apply>
<cn>-1</cn>
</apply>
notanumber)
notanumber represents the result of an
ill-defined floating point operation, sometimes also called
NaN.
<apply> <eq/>
<apply>
<divide/>
<cn>0</cn>
<cn>0</cn>
</apply>
<notanumber/>
</apply>
true)
true represents the logical constant for truth.
<apply> <eq/>
<apply>
<or/>
<true/>
<ci type = "logical">P</ci>
</apply>
<true/>
</apply>
false)
false represents the logical constant for falsehood.
<apply> <eq/>
<apply>
<and/>
<false/>
<ci type = "logical">P</ci>
</apply>
<false/>
</apply>
emptyset)
emptyset represents the empty set.
<apply>
<neq/>
<integers/>
<emptyset/>
</apply>
pi)
pi represents the mathematical constant which is
the ratio of a circle's circumference to its diameter, approximately
3.141592653.
<apply>
<approx/>
<pi/>
<cn type = "rational">22<sep/>7</cn>
</apply>
eulergamma)
eulergamma represents Euler's constant,
approximately 0.5772156649
<eulergamma/>
infinity)
infinity represents the concept of
infinity. Proper interpretation depends on context.
<infinity/>
Overview: Mathematical Markup Language (MathML) Version 2.0
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