Discrete Mathematics with Graph Theory (3rd Edition) 48

Discrete Mathematics with Graph Theory (3rd Edition) 48 -...

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46 Solutions to Exercises 21. (a) [BB] The ordered pairs defined by'" are (1,1), (1,4), (1,9), (2,2), (2,8), (3,3), (4,1), (4,4), (4,9), (5,5), (6,6), (7,7), (8,2), (8,8), (9,1), (9,4), (9,9). (b) [BB] I = {1,4,9} = 4 = 9;2 = {2,8} = 8; '3 = {3};"5 = {5}; 6 = {6}; '7 = {7}. (c) [BB] Since the sets {1, 4, 9}, {2, 8}, {3}, {5}, {6} and {7} partition A, they determine an equiv- alence relation, namely, that equivalence relation in which a '" b if and only if a and b belong to the same one of these sets. This is the given relation. 22. [BB] Reflexive: If a E A, then a 2 is a perfect square, so a '" a. Symmetric: If a '" b, then ab is a perfect square. Since ba = ab, ba is also a perfect square, so b '" a. Transitive: If a '" b and b '" e, then ab and be are each perfect squares. Thus ab = x 2 and be = y2 x 2 y2 (xy)2 for integers x and y. Now ab 2 e = x 2 y2, so ae = ~ = b . Because ae is an integer, so also x: is an integer. Therefore, a '" e. 23. (a) The order pairs
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