Editing Function (mathematics) (section)

=== Standard functions ===
There are a number of standard functions that occur frequently:
* For every set {{mvar|X}}, there is a unique function, called the '''{{vanchor|empty function}}''', or '''empty map''', from the [[empty set]] to {{mvar|X}}. The graph of an empty function is the empty set.<ref group=note>By definition, the graph of the empty function to {{mvar|X}} is a subset of the Cartesian product {{math|∅ × ''X''}}, and this product is empty.</ref> The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. Under the usual set-theoretic definition of a function as an [[Tuple|ordered triplet]] (or equivalent ones), there is exactly one empty function for each set, thus the empty function <math>\varnothing \to X</math> is not equal to <math>\varnothing \to Y</math> if and only if <math>X\ne Y</math>, although their graphs are both the [[empty set]].
* For every set {{mvar|X}} and every [[singleton set]] {{math|{{mset|''s''}}}}, there is a unique function from {{mvar|X}} to {{math|{{mset|''s''}}}}, which maps every element of {{mvar|X}} to {{mvar|s}}. This is a surjection (see below) unless {{mvar|X}} is the empty set.
* Given a function <math>f: X\to Y,</math> the ''canonical surjection'' of {{mvar|f}} onto its image <math>f(X)=\{f(x)\mid x\in X\}</math> is the function from {{mvar|X}} to {{math|''f''(''X'')}} that maps {{mvar|x}} to {{math|''f''(''x'')}}.
* For every [[subset]] {{mvar|A}} of a set {{mvar|X}}, the [[inclusion map]] of {{mvar|A}} into {{mvar|X}} is the injective (see below) function that maps every element of {{mvar|A}} to itself.
* The [[identity function]] on a set {{mvar|X}}, often denoted by {{math|id<sub>''X''</sub>}}, is the inclusion of {{mvar|X}} into itself.