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hw21 - Let n ∈ N n ≥ 2 and let i,j ∈ I n Assume that...

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Intro to Abstract Math Homework 21 Fall 2009 Due November 2 Exercise 60. Prove that for all n N , f : I n I n is an injection if and only if it is a surjection. Exercise 61. Let α and β denote the elements of S 3 with two-row representations ± 1 2 3 2 3 1 ² and ± 1 2 3 2 1 3 ² , respectively. Prove that every element of S 3 can be obtained as a combination of α and β . Exercise 62.
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Unformatted text preview: Let n ∈ N , n ≥ 2, and let i,j ∈ I n . Assume that i 6 = j . Define T : I n → I n by T ( x ) = x if x 6 = i and x 6 = j, j if x = i, i if x = j. a. Prove that T ∈ S n . The function T is called a transposition . b. What is the two-row description of T ?...
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