Content MathML: Quick List

The specification of MathML seems to contain two distinct languages: Presentation MathML and Content MathML. Not to put too fine a point on it, Presentation MathML is intended for marking up the look of expressions, while Content MathML is intended for marking up the meaning of expressions.

This page is a handy list of the elements in the latter. The list is based on MathML 2.0 elements. It also includes new elements in MathML 3.0. The deprecated elements ( reln, fn, declare ) are excluded.

Outline of this page

XContent MathML: Quick List
  1. Structure of Expressions
  2. Functions and Inverses
  3. Logic
  4. Set theory
  5. Arithmetic and Algebra
  6. Relations
  7. Sequences and Series
  8. Calculus and Vector calculus
  9. Linear algebra
  10. Statistics
  11. Transcendental functions
  12. Constants and Symbols
  13. Semantic mappings
  14. References

Structure of Expressions

Structure of Expressions
Name Remark Usage
cn number
  • 1.414]]>
  • A]]>
ci identifier
  • x]]>
  • v]]>
csymbol externally-defined symbol
  • Omega]]>
  • plus]]>
cs MathML 3.0string literal
  • <set> ... AB ]]>
cbytes MathML 3.0base64-encoded byte stream
cerror MathML 3.0to define a symbol that indicates an error (such as "division by zero")
  • <cerror> ... DivisionByZero divide x 0 ]]>
sep separates a string. used to mark up e-notations, rational numbers, complex numbers, etc.
  • 1.235]]>
  • 19]]>
  • 47]]>
apply apply a function
  • xy]]>
  • <apply><and/> ... 0t t1 ]]>
bind MathML 3.0binding
  • <bind><exists/> ... x xy x ]]>
bvar bound variables
  • <apply><forall/> ... a b a b ab ba ]]>
interval intervals. left/right - open/closed
  • <interval closure="open-closed"> ... 0 1 ]]>
  • <apply><int/> ... -11 f ]]>
condition a boolean valued expression. e.g., to use with logical quantifiers, to define sets, etc.
  • <interval> ... x 0x ]]>
  • <condition> ... x5 3 ]]>
  • <apply><int/> ... x xC fx ]]>
domainofapplication a domain of application. more general than interval or condition.
  • <apply><int/> ... A B f ]]>
share MathML 3.0to share and reuse XML sub trees
  • <apply><ci>f</ci> ...f f f a a ]]>

Functions and Inverses

Functions and Inverses
Name Remark Usage
lambda user-defined functions with concepts of lambda calculus
  • <lambda> ... x x1 ]]>
compose function composition operator
  • fgh]]>
  • <apply><eq/> ... fg x fgx ]]>
ident identity function
  • <apply><eq/> ... f f ]]>
inverse inverse function
  • <apply> ... f ]]>
  • <apply> ... A a ]]>
domain the domain of the function
  • <apply><eq/> ... f ]]>
codomain the codomain of the function
  • <apply><eq/> ... f ]]>
image the image of the function
  • <apply><eq/> ... -11 ]]>
piecewise piecewise function
  • <piecewise> ... x x0 0 x0 x x0 ]]>
piece a piece in piecewise function
otherwise "otherwise" piece
  • <piecewise> ... 0 x0 1 x1 x ]]>

Logic

Logic
Name Remark Usage
and boolean "and"
  • ab]]>
  • <apply><and/> ... i 0 n ai 0 ]]>
or boolean "or"
  • ab]]>
xor boolean "exclusive or"
  • ab]]>
not boolean "not"
  • a]]>
implies boolean relational operator "implies"
  • AB]]>
equivalent logically equivalent
  • <apply><equivalent/> ... a a ]]>
forall universal quantifier
  • <bind><forall/> ... x xx 0 ]]>
  • <bind><forall/> ... p q p q pq p q2 ]]>
exists existential quantifier
  • <bind><exists/> ... x fx 0 ]]>
  • <apply><exists/> ... x fx 0 ]]>

Set theory

Set theory
Name Remark Usage
set a set
  • <set> ... abc ]]>
  • <set> ... x x5 x ]]>
  • <set> ... S ST S ]]>
  • <set> ... x x5 x x ]]>
list a list
  • <list> ... abc ]]>
  • <list order="numeric"> ... x x5 ]]>
in membership relation. denote that the given element belongs to the given set.
  • aA]]>
notin negated membership relation.
  • aA]]>
subset subset relation. denote that the first is a subset of the second.
  • <apply><subset/> ... A B ]]>
notsubset negated subset relation. denote that the first is NOT a subset of the second.
  • <apply><notsubset/> ... A B ]]>
prsubset proper subset relation. denote that the first is a proper subset of the second.
  • <apply><prsubset/> ... A B ]]>
notprsubset negated proper subset relation. denote that the first is NOT a proper subset of the second.
  • <apply><notprsubset/> ... A B ]]>
union union of sets
  • AB]]>
  • <apply><union/> ... S L S ]]>
intersect intersection of sets
  • <apply><intersect/> ... A B ]]>
  • <apply><intersect/> ... S L S ]]>
setdiff set difference operator
  • <apply><setdiff/> ... A B ]]>
cartesianproduct Cartesian product operator
  • AB]]>
card cardinality function
  • <apply><eq/> ... A 5 ]]>

Arithmetic and Algebra

Arithmetic and Algebra
Name Remark Usage
plus addition
  • xyz]]>
minus unary minus or subtraction
  • 3]]>
  • xy]]>
times multiplication
  • ab]]>
divide division
  • <apply><divide/> ... a b ]]>
quotient quotient
  • ab]]>
rem remainder
  • ab]]>
power exponentiation
  • x3]]>
root n-th root, specified by degree
  • x]]>
  • <apply><root/> ... n a ]]>
factorial factorial
  • n]]>
gcd greatest common divisor
  • abc]]>
lcm lowest common multiple
  • abc]]>
abs absolute Value
  • x]]>
conjugate complex conjugate
  • <apply><conjugate/> ... x ⅈy ]]>
real the real part of a complex number
  • <apply><real/> ... x y ]]>
imaginary the imaginary part of a complex number
  • <apply><imaginary/> ... x y ]]>
arg the argument of a complex number
  • <apply><arg/> ... x y ]]>
floor round-down (towards negative infinity)
  • a]]>
ceiling round-up (towards positive infinity)
  • a]]>
max maximum
  • 235]]>
  • <apply><max/> ... y y 01 y3 ]]>
min minimum
  • ab]]>
  • <apply><min/> ... x xB x2 ]]>

Relations

Relations
Name Remark Usage
eq equal to
  • <apply><eq/> ... 24 12 ]]>
neq not equal to
  • 34]]>
gt greater than
  • 32]]>
lt less than
  • 234]]>
geq greater than or equal to
  • 433]]>
leq less than or equal to
  • 334]]>
approx approximately equal
  • <apply><approx/> ... 227 ]]>
factorof
  • ab]]>

Sequences and Series

Sequences and Series
Name Remark Usage
sum summation
  • <apply><sum/> ... i 0 100 xi ]]>
product product
  • <apply><product/> ... x a b f x ]]>
limit a limit of a sequence
  • <apply> ... x 0 x ]]>
  • <apply><limit/> ... x x0 x ]]>
tendsto tending to a specified value. "as x tends to y". used with limit. a binary relation.
  • <apply> ... x 2 a 2 ]]>
  • <apply> ... x y f x y g x y ]]>

Calculus and Vector calculus

Calculus and Vector calculus
Name Remark Usage
diff differentiation
  • <apply> ... f ]]>
  • <apply> ... x f x ]]>
partialdiff partial differentiation
  • <apply><partialdiff/> ... xm yn k f x y ]]>
  • <apply><partialdiff/> ... x y f x y ]]>
  • <apply><partialdiff/> ... 113 f ]]>
degree "degree" or "order" of an operation. often used in derivatives and moments.
  • <apply> ... x n y m x y ]]>
  • <apply> ... n a ]]>
  • <apply> ... 3 X ]]>
int definite or indefinite integral
  • <apply> ... ]]>
  • <apply> ... a b ]]>
lowlimit lower limit
  • <apply> ... x 0 x ]]>
uplimit upper limit
  • <apply> ... x 0 a f x ]]>
grad vector calculus gradient
  • <apply> ... f ]]>
curl vector calculus curl
  • <apply> ... a ]]>
divergence vector calculus divergence
  • <apply> ... a ]]>
laplacian vector calculus laplacian
  • <apply> ... f f ]]>

Linear algebra

Linear algebra
Name Remark Usage
vector vector
  • <vector> ... 1 2 3 x ]]>
matrix matrix
  • <matrix> ... 010 001 100 ]]>
matrixrow a row of a matrix
  • <matrixrow> ... 1 2 3 x ]]>
determinant determinant of a matrix
  • <apply> ... A ]]>
transpose transpose of a matrix
  • <apply> ... A ]]>
selector operator for indexing into vectors, matrices, and lists
  • <apply> ... A 3 2 ]]>
vectorproduct vector product of two vectors
  • <apply> ... A B a b θ N ]]>
scalarproduct scalar product of two vectors
  • <apply> ... A B a b θ ]]>
outerproduct outer product of two vectors
  • <apply> ... A B ]]>

Statistics

Statistics
Name Remark Usage
mean mean
  • <apply> ... X ]]>
sdev standard deviation
  • <apply> ... X ]]>
variance variance
  • <apply> ... X ]]>
median median
  • <apply> ... X ]]>
mode mode
  • <apply> ... X ]]>
moment moment. use degree for the n in "n-th moment". use momentabout for the p in "moment about p".
  • <apply> ... 3 p X ]]>
momentabout the p in "moment about p".
  • <apply> ... 2 0 X ]]>

Transcendental functions

Transcendental functions
Name Remark Usage
exp exponential function
  • <apply> ... x ]]>
ln natural logarithm function
  • <apply> ... a ]]>
log a logarithm to a base given by logbase
  • <apply> ... 3 x ]]>
sin sine
  • <apply> ... x ]]>
  • <apply> ... x x 3 ]]>
cos cosine
tan tangent
sec secant
csc cosecant
cot cotangent
sinh hyperbolic sine
cosh hyperbolic cosine
tanh hyperbolic tangent
sech hyperbolic secant
csch hyperbolic cosecant
coth hyperbolic cotangent
arcsin inverse sine
arccos inverse cosine
arctan inverse tangent
arcsec inverse secant
arccsc inverse cosecant
arccot inverse cotangent
arcsinh inverse hyperbolic sine
arccosh inverse hyperbolic cosine
arctanh inverse hyperbolic tangent
arcsech inverse hyperbolic secant
arccsch inverse hyperbolic cosecant
arccoth inverse hyperbolic cotangent

Constants and Symbols

Constants and Symbols
Name Remark Usage
primes the set of all natural prime numbers
  • <apply> ... 17 ]]>
naturalnumbers the set of all natural numbers
  • <apply> ... 1729 ]]>
integers the set of all integers
  • <apply> ... 42 ]]>
rationals the set of all rational numbers
  • <apply> ... 227 ]]>
reals the set of all real numbers
  • <apply> ... 44.997 ]]>
complexes the set of all complex numbers
  • <apply> ... 1729 ]]>
imaginaryi imaginary unit, square roots of −1
  • <apply><eq/> ... 2 -1 ]]>
emptyset the empty set
  • <apply> ... ]]>
exponentiale the exponential base of the natural logarithms
  • <apply><eq/> ... 1 ]]>
pi the ratio of a circle's circumference to its diameter
  • <apply> ... 227 ]]>
eulergamma Euler's gamma
  • ]]>
infinity infinity
  • ]]>
true the logical constant for truth
  • <apply><eq/> ... P ]]>
false the logical constant for falsehood
  • <apply><eq/> ... P ]]>
notanumber the result of an ill-defined floating point operation, also called NaN
  • <apply><eq/> ... 0 0 ]]>

Semantic mappings

Semantic mappings
Name Remark Usage
semantics a semantic mapping
  • <semantics> ... 123 456 N[123/456, 39] $0.269736842105263157894736842105263157894\ldots$ evalf(123/456, 39); 0.269736842105263157894 736842105263157894 ‾ 123 456 ]]>
annotation a semantic annotation in a non-XML format
  • <semantics> ... x 5 \sin x + 5 ]]>
annotation-xml a semantic annotation in an XML format
  • <semantics> ... C2 C2 ]]>
  • <semantics> ... x 5 5 ]]>

References

More pragmatic in Probability and Statistics, see:

The language specifications:

Content dictionaries: