![]() ![]() In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. A permutation formed by using even transpositions is called an even permutation, and a permutation made by using odd transpositions is called an odd permutation.Mathematical version of an order change Each of the six rows is a different permutation of three distinct balls Recall that here, if f(x) and g(x) are two functions then f.g(x) is given by f(g(x)) and g.f(x) is given by g(f(x)). Multiplication of permutations is very similar to the formation of composite functions. If there are n numbers in a set, the number of elements in its permutation group equals n. The set of all the permutations of a set for which permutations are defined is called the permutation group of the set. If a set contains n elements, then we can say that the cardinal number of the set is n, and this number is also known as the degree of the permutation. Permutations of a set are bijective mappings (one-one onto mappings)on the set by itself. Since p1 is odd and I is even, so by property 1, the permutation p2 is also an even permutation. If p1 is an odd permutation and p2 is its inverse, then p1.p2=I is the identity permutation. Since p1 and I are both even so, by using property 1, we can conclude that the permutation p2 is also even.ĥ) Let p1 be an odd permutation, and then the inverse of p1 will also be an odd permutation. If p1 is an even permutation and p2 is its inverse permutation, then p1.p2=I is the identity permutation. The inverse of p1 will also be an even permutation. The identity permutation of a set can always be expressed as a product of two even permutations and thus, by the use of the first property, is always even.Ĥ) Let p1 be an even permutation. Hence the product p1.p2 is an odd permutation.ģ) The identity permutation is always even. Thus the product p1.p2 results from 2m+2n+1 transpositions, which is an odd number. ![]() Let us assume that only one of p1 or p2 is odd and thus is the result of 2m+1 transpositions while the other is the result of 2n transpositions. Thus we can conclude that the product p1.p2 is also even.Ģ) If p1 and p2 are permutations such that only one out of p1 and p2 is odd, then the product p1.p2 is odd. This means that the product p1.p2 results from 2a+2b transpositions which is an even number, so the product p1.p2 is also even.Ĭase 2: Given that p1 and p2 are two permutations of the same set or two different sets such that both of them are odd, then p1 and p2 are both results of 2a+1 and 2b+1 transpositions, respectively, and thus their product p1.p2 is the result of 2(a+b+1) transpositions which is also an even number. p 2 is even.Ĭase 1: Given that p1 and p2 are two permutations of the same set or two different sets such that both of them are even, then p1 and p2 are both results of 2a and 2b transpositions, respectively. In contrast, a permutation formed by using an even number of permutations is known as an even permutation.Ī look at the properties of even and odd permutations:ġ) Given that p 1 and p 2 are two permutations of different sets or the same set such that both p 1 and p 2 are either even or odd, then the product of these permutations p 1. Even and Odd PermutationsĪ permutation which can be formed using an odd number of transpositions is called an odd permutation. In A 5 goes to 5 and in B 5 goes to 1 so in A.B 5 goes to 1 and finally in A 1 goes to 1 and in B 1 goes to 4 so in A.B 1 goes to 3. In A 4 goes to 2 and in B 2 goes to 2 so in A.B 4 goes to 2. Similarly, in A 3 goes to 4 and in B 4 goes to 5 so in A.B 2 goes to 5. ![]() We can see that in the permutation A 2 goes to 3 and in B 4 goes to 4 thus, in A.B 2 goes to 4. Let us try to look into what is happening here. Let us assume that a set B has the following elements ![]() A permutation of a set is the one-one onto mapping of the set to itself, that is, bijective mapping. If there are n numbers in the original set, then its cardinal number is also n, and the number of elements in its permutation group equals n! Definition of Permutation GroupĪ permutation group of a set is another set which contains all possible permutations that can be created from the original set. The permutation group of a set can be defined as a set containing all the possible permutations that can be created for the original set. If the cardinal number (the total number of elements in a set) of a finite set is n, then n is also called the degree of permutation of the set. A bijective relation refers to one-one mapping. A permutation of a finite set is a bijective relation from itself to itself. A permutation of a set is defined for finite sets only. ![]()
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