Question

ath 3034 Friday, November 8 Ninth Homework Due 9:05 a.m., Friday November 15 1. Let be a binary operation on a set S with an identity e (necessarily unique). (a) Prove that e is invertible and has a unique inverse. (b) Let s ES{el. Prove that e is not an inverse for s. (c) Suppose that S2. Prove that inverses (if they exist) are unique for every element of S. (4 points) 2. (cf. Problem 7.3.5 on p. 179) Let a: AB, B: AB and y: BC be maps such that yo a yo B. Prove that aB, that is a(a)B(a) for every a E A. (3 points) 3. (cf. Problem 7.3.7 on p. 179) In each of the following, determine whether the fol- lowing maps are invertible. In the case the mapping is invertible, exhibit the inverse mapping. (a) f: R R defined by f(x)- 2x+5 (b) Y: M2(R) M2(R) defined by a(A) CA, where C (3 points) (3 problems, 10 points altogether)

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Answer 1

(a) ene-e [ie in an identity 1 Here e in invertible. euphore, there in another inverse of e, sorg a. Then, ande = a [:e is an identity] and, mirce, a in an invern of e. a= are in also an indentity So, ate = e [ na in an identity] have Jeram (i) and 6), aze. Hence, inverse of e in unique (t) If e be an innere of s, then sin an inverse of e. But as inrere of e in unigre, we must se. But sts-{e}. which given a conteraduction: so, & is not an inverse of s. (6) einn, 1s=8.2 s={e, a}. from a, etze and, from (6) at ts-{e} = {a} in, atza

Hence, incore of e in uniquely e 1 la 11 2 Auertian in warong. Pos he then, roa = roß but 2 (2)=526=3(2)