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Let A be the set of all permutations f : Z→Z such that there exists a positive integer n for which, for all ie Z, we have f(i+n) = f(i)+n. For example, for each k E Z, the “add k" function a : Z→ Z defined by a¿(i) =i+k lies in A. (a) Write down an example of an element of A that is not one of the functions az above. (b) Prove that A is a group with the operation of function composition.

Let A be the set of all permutations f : Z→Z such that there exists a positive integer n for which, for all ie Z, we have f(i+n) = f(i)+n. For example, for each k E Z, the “add k" function a : Z→ Z defined by a¿(i) =i+k lies in A. (a) Write down an example of an element of A that is not one of the functions az above. (b) Prove that A is a group with the operation of function composition.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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t Let A be the set of all permutations f : Z → Z such that there exists a positive integer n for which, for all i e Z, we have f(i+n) = f(i)+n. For example, for each k E Z, the “add k" function ag : Z→ Z defined by ar(i) =i+k lies in A. (a) Write down an example of an element of A that is not one of the functions ar above. (b) Prove that A is a group with the operation of function composition.
Transcribed Image Text:t Let A be the set of all permutations f : Z → Z such that there exists a positive integer n for which, for all i e Z, we have f(i+n) = f(i)+n. For example, for each k E Z, the “add k" function ag : Z→ Z defined by ar(i) =i+k lies in A. (a) Write down an example of an element of A that is not one of the functions ar above. (b) Prove that A is a group with the operation of function composition.
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