Let A be the set of all permutations f: Z Z such that there exists a positive integern for which, for all i e Z, we havef(i+n) = f(i) +n.For example, for each k E Z, the "add k" function a, : Z Z defined by a, (0-i+klies in A.(a) Write down an example of an element of A that is not one of the functions aabove.(b) Prove that A is a group with the operation of function composition.

Consider the given set A of all permutations f:ℤ→ℤ such that: f(i+n)=f(i)+n where n is a positive…

Answered: (a) Write down an example of an element…

(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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Question

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 a, (0-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.

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