Bijection - Inverses

Inverses

A bijection f with domain X ("functionally" indicated by f: X → Y) also defines a relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not usually yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y. Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. Bijections are the invertible functions.

Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition

for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position.