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Concept
Bijective proof
Summary
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.
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.
The key idea of the proof may be understood from a simple example: selecting k children to be rewarded with ice cream cones, out of a group of n children, has exactly the same effect as choosing instead the n − k children to be denied ice cream cones.
More abstractly and generally, the two quantities asserted to be equal count the subsets of size k and n − k, respectively, of any n-element set S. Let A be the set of all k-element subsets of S, the set A has size Let B be the set of all n−k subsets of S, the set B has size . There is a simple bijection between the two sets A and B: it associates every k-element subset (that is, a member of A) with its complement, which contains precisely the remaining n − k elements of S, and hence is a member of B. More formally, this can be written using functional notation as, f : A → B defined by f(X) = Xc for X any k-element subset of S and the complement taken in S. To show that f is a bijection, first assume that f(X1) = f(X2), that is to say, X1c = X2c. Take the complements of each side (in S), using the fact that the complement of a complement of a set is the original set, to obtain X1 = X2. This shows that f is one-to-one. Now take any n−k-element subset of S in B, say Y. Its complement in S, Yc, is a k-element subset, and so, an element of A.
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Related concepts (5)
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In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labeled tree is a unique sequence associated with the tree. The sequence for a tree on n vertices has length n − 2, and can be generated by a simple iterative algorithm. Prüfer sequences were first used by Heinz Prüfer to prove Cayley's formula in 1918. One can generate a labeled tree's Prüfer sequence by iteratively removing vertices from the tree until only two vertices remain. Specifically, consider a labeled tree T with vertices {1, 2, .
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In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which call "one of the most important tools in combinatorics", one describes a finite set from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other.
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