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Discrete Mathematics with Graph Theory (3rd Edition) 53

# Discrete Mathematics with Graph Theory (3rd Edition) 53 -...

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Chapter 2 51 10. This follows quickly from one of the laws of De Morgan and the identity X " Y = X n YC. A" (B n C) = An (B n C)C = An (B C U CC) ::l:: (A nBC) U (A n CC) = (A" B) U (A " C), using (3), p. 62 at the spot marked with the arrow. 11. (a) A binary relation on A is a subset of Ax A. (b) If A has 10 elements, A x A has 100 elements, so there are 2 100 binary relations on A. 12. Reflexive: For any a with lal ~ 1, we have a 2 = la 2 1 = lallal ~ lal, thus (a, a) E R. Symmetric by definition. Not antisymmetric because (1 1) is in R «(1)2 < 1 and (1)2 < 1) but 1 -'- 1 2' 4 2 - 4 4 - 2 2 ., Not transitive. We have (~, PER because (~)2 ~ 1 and (1)2 ~ ~ and (1, g) E R because (1)2 ~ g and (g)2 ~ 1, but (2' g) is not in R because (~)21, g. 13. (a) Reflexive: For any natural number a, we have a ~ 2a, so a '" a. Not symmetric: 2 '" 5 because 2 ~ 2(5), but 5 rf 2 because 5 1, 2(2). Not antisymmetric: Let a = 1 and b = 2. Then a '" b because 1 ~ 2(2) and also b '" a because 2 ~ 2(1). Not transitive: Let a = 3, b = 2, c = 1. Then a '" b because 3 ~ 2(2) and b '" c because 2 ~ 2(1). However, a rf c because 31, 2(1).
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