Bijection
For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function.
The elementary operation of counting establishes a bijection from some finite set to the first natural numbers (1, 2, 3, ...), up to the number of elements in the counted set.
It results that two finite sets have the same number of elements if and only if there exists a bijection between them.
More generally, two sets are said to have the same cardinal number if there exists a bijection between them.
Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations.
Galois correspondences are bijections between sets of mathematical objects of apparently very different nature.
For a binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold: Satisfying properties (1) and (2) means that a pairing is a function with domain X.
[3] Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up).
Property (2) is satisfied since no player bats in two (or more) positions in the order.
A group of students enter the room and the instructor asks them to be seated.
After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in.
What the instructor observed in order to reach this conclusion was that: The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.
A bijection f with domain X (indicated by f: X → Y in functional notation) also defines a converse relation starting in Y and going to X (by turning the arrows around).
Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition 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.
Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain.
[4] Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of A and B′ is a subset of B.
[6] An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.
[7] This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.
Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic may be found in any of these:
