Bijection

A bijection, or bijective function, is a function between two sets under which each element of the second set, the codomain, is the image of exactly one element of the first set, the domain, so the function pairs every element of either set with exactly one element of the other. A function is bijective exactly when it is invertible, meaning there exists an inverse function that undoes it in both directions; for example, multiplication by two defines a bijection from the integers to the even numbers, with division by two as its inverse. A function is bijective if and only if it is both injective, meaning no two elements of the domain share an image, and surjective, meaning every element of the codomain is reached. Counting itself is a bijection from a finite set to an initial segment of the natural numbers, so two finite sets have the same number of elements exactly when a bijection exists between them, and more generally two sets are said to share a cardinal number when a bijection exists between them; a bijective function from a set to itself is also called a permutation.