Please do Exerise 14.6.10 please show step by step and explain Now here's the punch line. We know that every permutation can be writ-ten as a product of transpositions. From what we have just shown, an oddpermutation must be the product of an odd number of transpositions; whilean even permutation must be the product of an even number of transposi-tions. It is impossible to write an even permutation as the product of an oddnumber of transpositions; and vice versa. We summarize our conclusions inthe following proposition.Proposition 14.6.9. A permutation o can be written as the product of aneven number of transpositions if and only if o is an even permutation. Also,o can be written as the product of an odd number of transpositions if andonly if o is an odd permutation.Exercise 14.6.10. Prove that it is impossible to write the identity permu-tation as the product of an odd number of transpositions.

Answered: Exercise 14.6.10. Prove that it is…

Exercise 14.6.10. Prove that it is impossible to write the identity permu- tation as the product of an odd number of transpositions.

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Counting And Probability

Section9.1: Counting

Problem 4E: True or False? In counting combinations, order matters. In counting permutations, order matters. For...

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Question

Please do Exerise 14.6.10 please show step by step and explain

Now here's the punch line. We know that every permutation can be writ-
ten as a product of transpositions. From what we have just shown, an odd
permutation must be the product of an odd number of transpositions; while
an even permutation must be the product of an even number of transposi-
tions. It is impossible to write an even permutation as the product of an odd
number of transpositions; and vice versa. We summarize our conclusions in
the following proposition.
Proposition 14.6.9. A permutation o can be written as the product of an
even number of transpositions if and only if o is an even permutation. Also,
o can be written as the product of an odd number of transpositions if and
only if o is an odd permutation.
Exercise 14.6.10. Prove that it is impossible to write the identity permu-
tation as the product of an odd number of transpositions.

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