A bijection (or bijective function) is a mathematical function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto.

In simple terms, a bijective function creates a one-to-one correspondence between its possible input values and possible output values. (In some references, the phrase "one-to-one" is used alone to mean bijective. Wikipedia does not follow this older usage.)

More formally, a function fX → Y is bijective if for every y in the codomain Y there is exactly one x in the domain X with f(x) = y.

Surjective, not injective

Injective, not surjective


Not surjective, not injective

When X and Y are both the real line R, then a bijective function fR → R can be visualized as one whose graph is intersected exactly once by any horizontal line.

If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Generalising this to infinite sets leads to the concept of cardinal number, a way to distinguish the various infinite sizes of infinite sets.

Examples and counterexamples

Consider the function fR → R defined by f(x) = 2x + 1. This function is bijective, since given an arbitrary real number y, we can solve y = 2x + 1 to get exactly one real solution x = (y − 1)/2.

On the other hand, the function gR → R defined by g(x) = x2 is not bijective, for two essentially different reasons. First, we have (for example) g(1) = 1 = g(−1), so that g is not injective; also, there is (for example) no real number x such that x2 = −1, so that g is not surjective either. Either one of these facts is enough to show that g is not bijective.

However, if we define the function hR+ → R+ by the same formula as g, but with the domain and codomain both restricted to only the nonnegative real numbers, then the function h is bijective. This is because, given an arbitrary nonnegative real number y, we can solve y = x2 to get exactly one nonnegative real solution x = √y.


See also: Injective function, Surjection