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(a) (12537)-1 (c) [(1235)(467)]-2

(a) (12537)-1 (c) [(1235)(467)]-2

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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Please do Exercise 14.4.34 part A and C

PROOF. By Proposition 14.4.26 we can write µ = (a,a,)(a,an-1)-.. (a,a3)(aja2). Now consider first just the last two transpositions in this expression. In the Functions chapter, we proved the formula (fog) = gof for invertible functions f and g. Since transpositions are invertible functions, we have (laras) (a102) = (a1a2)-"(a1a3)-| = (a1a2)(a1a3) (the second equality follows because every transposition is its own inverse.) If we apply similar reasoning to the last three transpositions in the ex- pression, we find = [(a,a3)(a,a2)](aja4)- = (a,a2)(aja3)(a,a4) Applying this result inductively, we obtain finally: = (a,a2) (a1as) (a1an-1)(a1an), from this expression we may see that a1 + an, an + an-1, an-1+ an-2, - .., a2 + a1, which corresponds to the cycle we want. Because the product of permutations is an associative operation, we may find the inverse of any product of cycles by taking the inverses of the cycles in reverse order. (Actually, this is just a special case of the inverse of function composition: (fi o f2 0.. o fn-1° fn)¬ = f o fn0...o f2 o fi.) Example 14.4.32. [(1498)(2468)]- = (2468)-'(1498)-1 = (2864)(1894) = (164)(289). Example 14.4.33. (1357)-2 = [(1357)-)² = (1753)² = (1753)(1753) : (15)(37). %3D Exercise 14.4.34. Calculate each of the following. (a) (12537)-1 (c) [(1235)(467)]-2 (b) [(12)(34)(12)(47)|~| (d) (1254)-(123)(45)(1254)
Transcribed Image Text:PROOF. By Proposition 14.4.26 we can write µ = (a,a,)(a,an-1)-.. (a,a3)(aja2). Now consider first just the last two transpositions in this expression. In the Functions chapter, we proved the formula (fog) = gof for invertible functions f and g. Since transpositions are invertible functions, we have (laras) (a102) = (a1a2)-"(a1a3)-| = (a1a2)(a1a3) (the second equality follows because every transposition is its own inverse.) If we apply similar reasoning to the last three transpositions in the ex- pression, we find = [(a,a3)(a,a2)](aja4)- = (a,a2)(aja3)(a,a4) Applying this result inductively, we obtain finally: = (a,a2) (a1as) (a1an-1)(a1an), from this expression we may see that a1 + an, an + an-1, an-1+ an-2, - .., a2 + a1, which corresponds to the cycle we want. Because the product of permutations is an associative operation, we may find the inverse of any product of cycles by taking the inverses of the cycles in reverse order. (Actually, this is just a special case of the inverse of function composition: (fi o f2 0.. o fn-1° fn)¬ = f o fn0...o f2 o fi.) Example 14.4.32. [(1498)(2468)]- = (2468)-'(1498)-1 = (2864)(1894) = (164)(289). Example 14.4.33. (1357)-2 = [(1357)-)² = (1753)² = (1753)(1753) : (15)(37). %3D Exercise 14.4.34. Calculate each of the following. (a) (12537)-1 (c) [(1235)(467)]-2 (b) [(12)(34)(12)(47)|~| (d) (1254)-(123)(45)(1254)
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