By Carol Whitehead (auth.)

ISBN-10: 0333426576

ISBN-13: 9780333426579

ISBN-10: 1349090417

ISBN-13: 9781349090419

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**Example text**

Fig. 2 2. Let S = {1, 2, 3, 4} and R 2 be defined on S by a R2 b ~ a + b :::; 5. The set of arcs is given by A = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}. (See Fig. ) 4 Fig. 3 51 GUIDE TO ABSTRACT ALGEBRA 3. Let S = {1, 2, 3, 4, 5, 6} and R 3 be defined on S by a R3 b 4 ¢> a = 2b. 2 ~ 6 • 5 3 Fig. 4 The set of arcs is given by A = {(2, 1), (4, 2), (6, 3)}. (See Fig. 5a I Draw graphs to illustrate the following relations on the set s ={1, 2, 3, 4, 5, 6}.

Hence from (i) and (ii), ii = b. • It is easy to show that the converse of this result also holds. 2 a = b =>a ~ b, 'I a, b E S. Proof Suppose Zi =b. We know a E a, by the reflexive property of Hence a E b, since a= b. But a E b => a ~ b, by definition of b. • ~. 2 gives that for any equivalence relation ~ on any set S, This result justifies our earlier assumption that any element of an equivalence class can equally well serve as class representative. Note that we 42 RELATIONS have had to use all three properties of an equivalence relation in the proof.

B = 0 =>A B = A s B'. Is B sA' also true? n B'. 5 Prove that A' n B' = (A U B)'. Hence show that A' n (B U C)' = (A U B)' n C'. 6 Prove that (A' n B)'. (b) A (a) (A')' =A; Hence prove that (A' n B)' U C = A U C if and only if B' sA U C. ) s 7 Prove that (A n B) U (B n C) U (C n A) = (A U B) n (B U C) n (C U A). What identity is obtained by putting C = 0 in this result? 4b In the following problems, A, B, C, D are subsets of a set 'fo. 1 Put the correct sign =>, <= or ditions on a real number x: ¢> between the following pairs of con- 1 (a) x E Z+;- E 0.

### Guide to Abstract Algebra by Carol Whitehead (auth.)

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