Discrete Mathematics with Graph Theory (3rd Edition) 51

Discrete Mathematics with Graph Theory (3rd Edition) 51 -...

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Section 2.5 (b) Assuming it exists, the least upper bound of A and B has two properties: (1) A ~ L, B ~ L; (2) if A ~ C and B ~ C, then L ~ C. 49 We must prove that Au B has these properties. Since A ~ Au B and B ~ AU B, AU B satisfies (1). Also, if A ~ C and B ~ C, then A U B ~ C, so A U B satisfies (2) and A U B = A VB. 13. (a) [BB] a V b = b and here is why. We are given a :5 b and have b :5 b by reflexivity. Thus b is an upper bound for a and b. It is least because if c is any other upper bound, then a :5 c, b :5 c; in particular, b :5 c. (b) a 1\ b = a and here is why. We are given a :5 b and have a :5 a by reflexivity. Thus a is a lower bound for a and b. It is greatest because if c is any other lower bound, then c :5 a, c :5 b; in particular, c :5 a. 14. (a) [BB] Suppose x and y are each glbs of two elements a and b. Then x :5 a, x :5 b implies x :5 y because y is a greatest lower bound, and y :5 -a, y :5 b implies y :5 x because x is greatest. So, by antisymmetry, x = y. (b) Suppose
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