MATH 2272
(Lecturer: Mr. A. Daaga)
Semester II (2016-2017)
Assignment #1 The due date for this assignment is Friday, February 3rd , 2017, before 9:00am. Selected questions will be marked. 1. A relation R is defined on a set S = {a, b, c, d} by the following subsets of S: (a) = {a, b, c}, (b) = {a, b, c}, (c) = {a, b} and (d) = {d}, where (x) denotes the set of elements that are related to x. Is R reflexive, symmetric or transitive? Give reasons for your answer.
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2. Construct the operation table for the group (Zn , +). # 3. Let Z# n be the set Zn − {[0]}. Determine whether or not Z6 is a group under the operation (defined by [a] [b] = [ab]). [2]
4. Which of the following sets are groups under the indicated operations? Give reasons for your answer. Note that Z, Q, and C denote the sets of integers, rational numbers and complex numbers respectively. (a) Z under the operation ? defined by a ? b = a − b.
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(b) Q − {1} under the operation ? defined by a ? b = a + b − ab.
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(c) {a + bi ∈ C : a and b are integers}, where the operation is addition. √ 5. Let S = {a + b 2 : a, b ∈ Q and a and b are not simultaneously zero}. (a) Show that S is a group under the usual multiplication in R.
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(b) Could you define another multiplication under which S is a group? 6. If a and b are in G and ab = ba, we say that a and b commute. Assuming that a and b commute, prove the following: (a) a2 commutes with b2 .
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(b) (ab) = an bn for every positive integer n [Proof by induction].
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7. Let G be a group. Let a, b, c denote elements of G, and let e be the identity element of G. Prove the following: (a) If abc = e, then cab = e and bca = e.
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(b) For every positive integer n,(bab−1 )n = ban b−1 .
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