In combinatorics, bijective proof is a proof technique that finds a bijective function between two finite sets and, or a size-preserving bijective function between two combinatorial classes, thus proving that they have the same number of elements,. One place the technique is useful is where we wish to know the size of, but can find no direct way of counting its elements. By establishing a bijection from to some solves the problem if 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.
The symmetry of the binomial coefficients states that This means that there are exactly as many combinations of things in a set of size as there are combinations of things in a set of size .
A bijective proof
The key idea of the proof may be understood from a simple example: selecting out of a group of children which to reward with ice cream cones has exactly the same effect as choosing instead the children to be denied them. More abstractly and generally, the two quantities asserted to be equal count the subsets of size and, respectively, of any -element set. Let be the set of all -element subsets of, the set has size Let be the set of all subsets of, the set B has size. There is a simple bijection between the two sets and : it associates every -element subset with its complement, which contains precisely the remaining elements of, and hence is a member of. More formally, this can be written using functional notation as, defined by for any -element subset of and the complement taken in. To show that f is a bijection, first assume that, that is to say,. Take the complements of each side, using the fact that the complement of a complement of a set is the original set, to obtain. This shows that is one-to-one. Now take any -element subset of in, say. Its complement in,, is a -element subset, and so, an element of. Since, 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,.
