Bijective Function

A bijective function is a function that is both a (One-to-One) Injective Function and a Surjective Function.

• AKA: Bijection.
• See: Bijective Relation, Graph Isomorphism(Edge Preserving Bijection).

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References

2011

• (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Bijection
  • In mathematics, a bijection, or a bijective function, is a function [math]\displaystyle{ f }[/math] from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that [math]\displaystyle{ f }[/math](x) = y. It follows from this definition that no unmapped element exists in either X or Y. Alternatively, [math]\displaystyle{ f }[/math] is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective). For example, consider the function succ, defined from the set of integers [math]\displaystyle{ \scriptstyle\Z }[/math] to [math]\displaystyle{ \scriptstyle\Z }[/math], that to each integer x associates the integer succ(x) = x + 1. For another example, consider the function sumdif that to each pair (x,y) of real numbers associates the pair sumdif(x,y) = (x + y, x − y). A bijective function from a set to itself is also called a permutation. The set of all bijections from X to Y is denoted as X ↔ Y. (Sometimes this notation is reserved for binary relations, and bijections are denoted by X ⤖ Y instead.) Occasionally, the set of permutations of a single set X may be denoted X!. Bijective functions play a fundamental role in many areas of mathematics, for instance in the definition of isomorphism (and related concepts such as homeomorphism and diffeomorphism), permutation group, projective map, and many others.

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