To submit 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 az : 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 a above. (b) Prove that A is a group with the operation of function composition.
To submit 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 az : 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 a above. (b) Prove that A is a group with the operation of function composition.
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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Transcribed Image Text:To submit 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 a : Z →Z defined by ax(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.
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