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(This ex 14 For n 24 how many elements in Sn are there that map 1 to 2, 2 to 3, and 3 to 1? For n ≥ 5, how many are in An?

(This ex 14 For n 24 how many elements in Sn are there that map 1 to 2, 2 to 3, and 3 to 1? For n ≥ 5, how many are in An?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 14

Sure, here's the transcription: --- 12. If \( a \) is even, prove that \( a^n \) is even. If \( a \) is odd, prove that \( a^n \) is odd if and only if \( n \) is odd. 13. Show that a function from a finite set \( S \) to itself is one-to-one if and only if it is onto. Is this true when \( S \) is infinite? (This exercise is referred to in Chapter 6.) 14. For \( n \geq 4 \) how many elements in \( S_n \) are there that map 1 to 2, 2 to 3, and 3 to 1? For \( n \geq 5 \), how many are in \( A_n \)? 15. Let \( \alpha \) and \( \beta \) belong to \( S_n \). Prove that \( \alpha \beta \) is even if and only if \( \alpha \) and \( \beta \) are both even or both odd. 16. Associate an even permutation with the number \( +1 \) and an odd permutation with the number \( -1 \). Draw an analogy between the result of multiplying two permutations and the result of multiplying their corresponding numbers \( +1 \) or \( -1 \). Complete the following statement: ___. --- There are no graphs or diagrams in the image.
Transcribed Image Text:Sure, here's the transcription: --- 12. If \( a \) is even, prove that \( a^n \) is even. If \( a \) is odd, prove that \( a^n \) is odd if and only if \( n \) is odd. 13. Show that a function from a finite set \( S \) to itself is one-to-one if and only if it is onto. Is this true when \( S \) is infinite? (This exercise is referred to in Chapter 6.) 14. For \( n \geq 4 \) how many elements in \( S_n \) are there that map 1 to 2, 2 to 3, and 3 to 1? For \( n \geq 5 \), how many are in \( A_n \)? 15. Let \( \alpha \) and \( \beta \) belong to \( S_n \). Prove that \( \alpha \beta \) is even if and only if \( \alpha \) and \( \beta \) are both even or both odd. 16. Associate an even permutation with the number \( +1 \) and an odd permutation with the number \( -1 \). Draw an analogy between the result of multiplying two permutations and the result of multiplying their corresponding numbers \( +1 \) or \( -1 \). Complete the following statement: ___. --- There are no graphs or diagrams in the image.
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