# Bijective proof

In combinatorics, bijective proof is a proof technique that finds a bijective function (that is, a one-to-one and onto function) f : A → B between two finite sets A and B, or a size-preserving bijective function between two combinatorial classes, thus proving that they have the same number of elements, |A| = |B|.

One place the technique is useful is where we wish to know the size of A, but can find no direct way of counting its elements.

By establishing a bijection from A to some B solves the problem if B is more easily countable.

Another useful feature of the technique is that the nature of the bijection itself often provides powerful insights into each or both of the sets.

## Basic examples

### Proving the symmetry of the binomial coefficients

The symmetry of the binomial coefficients states that

This means that there are exactly as many combinations of k things in a set of size n as there are combinations of n − k things in a set of size n.

#### A bijective proof

The key idea of the proof may be understood from a simple example: selecting out of a group of n children which k to reward with ice cream cones has exactly the same effect as choosing instead the n − k children to be denied them.

## Other examples

Problems that admit bijective proofs are not limited to binomial coefficient identities.

As the complexity of the problem increases, a bijective proof can become very sophisticated.

This technique is particularly useful in areas of discrete mathematics such as combinatorics, graph theory, and number theory.

The most classical examples of bijective proofs in combinatorics include:

- Prüfer sequence, giving a proof of Cayley's formula for the number of labeled trees.
- Robinson-Schensted algorithm, giving a proof of Burnside's formula for the symmetric group.
- Conjugation of Young diagrams, giving a proof of a classical result on the number of certain integer partitions.
- Bijective proofs of the pentagonal number theorem.
- Bijective proofs of the formula for the Catalan numbers.

## See also

- Binomial theorem
- Schröder–Bernstein theorem
- Double counting (proof technique)
- Combinatorial principles
- Combinatorial proof
- Categorification

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Bijective proof.