Английская Википедия:Bijective proof
Шаблон:Short description In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other. This technique can be useful as a way of finding a formula for the number of elements of certain sets, by corresponding them with other sets that are easier to count. Additionally, 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
- <math> {n \choose k} = {n \choose n-k}. </math>
This means that there are exactly as many combinations of Шаблон:Math things in a set of size Шаблон:Math as there are combinations of Шаблон:Math things in a set of size Шаблон:Math.
A bijective proof
The key idea of the proof may be understood from a simple example: selecting Шаблон:Math children to be rewarded with ice cream cones, out of a group of Шаблон:Math children, has exactly the same effect as choosing instead the Шаблон:Math children to be denied ice cream cones.
More abstractly and generally,[1] the two quantities asserted to be equal count the subsets of size Шаблон:Math and Шаблон:Math, respectively, of any Шаблон:Math-element set Шаблон:Math. Let Шаблон:Mvar be the set of all Шаблон:Mvar-element subsets of Шаблон:Mvar, the set Шаблон:Mvar has size <math>\tbinom{n}{k}.</math> Let Шаблон:Mvar be the set of all Шаблон:Mvar subsets of Шаблон:Mvar, the set B has size <math>\tbinom{n}{n-k}</math>. There is a simple bijection between the two sets Шаблон:Mvar and Шаблон:Mvar: it associates every Шаблон:Math-element subset (that is, a member of Шаблон:Mvar) with its complement, which contains precisely the remaining Шаблон:Math elements of Шаблон:Math, and hence is a member of Шаблон:Mvar. More formally, this can be written using functional notation as, Шаблон:Math defined by Шаблон:Math for Шаблон:Mvar any Шаблон:Mvar-element subset of Шаблон:Mvar and the complement taken in Шаблон:Mvar. To show that f is a bijection, first assume that Шаблон:Math, that is to say, Шаблон:Math. Take the complements of each side (in Шаблон:Mvar), using the fact that the complement of a complement of a set is the original set, to obtain Шаблон:Math. This shows that Шаблон:Mvar is one-to-one. Now take any Шаблон:Mvar-element subset of Шаблон:Mvar in Шаблон:Mvar, say Шаблон:Mvar. Its complement in Шаблон:Mvar, Шаблон:Math, is a Шаблон:Mvar-element subset, and so, an element of Шаблон:Mvar. Since Шаблон:Math, Шаблон:Mvar is also onto and thus a bijection. The result now follows since the existence of a bijection between these finite sets shows that they have the same size, that is, <math>\tbinom{n}{k} = \tbinom{n}{n-k}</math>.
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
References
Further reading
- Loehr, Nicholas A. (2011). Bijective Combinatorics. CRC Press. Шаблон:ISBN, Шаблон:ISBN.
External links
- "Division by three" – by Doyle and Conway.
- "A direct bijective proof of the hook-length formula" – by Novelli, Pak and Stoyanovsky.
- "Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees" – by Gilles Schaeffer.
- "Kathy O'Hara's Constructive Proof of the Unimodality of the Gaussian Polynomials" – by Doron Zeilberger.
- "Partition Bijections, a Survey" – by Igor Pak.
- Garsia-Milne Involution Principle – from MathWorld.
