Discrete Mathematics with Graph Theory (3rd Edition) 37

Discrete Mathematics with Graph Theory (3rd Edition) 37 -...

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Section 2.2 35 In this case, b E A EEl B, so bE A EEl C and since b fJ. A it follows that bE C. Case 2: b E A. Here we have b E B n A and, hence, b fJ. A EEl B, so b fJ. A EEl C. Since b E A, we must have bE C (otherwise, bE A, C <;;; A EEl C). In either case, we obtain bE C. It follows that B <;;; C. A similar argument shows C <;;; Band, hence, C = B. (d) This is false since for A = 0, A x B = A x C = 0 regardless of Band C. 28. (a) True. Let (a, b) E A x B. Since a E A and A <;;; C, we have a E C. Since b E Band B <;;; D, bED. Thus, ( a, b) E C x D and A x B <;;; C x D. (b) False: Consider A = {I}, B = {2, 3}, C = {I, 2, 3}. (c) False. Let A = {I}, B = 0, C = {2}, D = {3}. Then A x B = 0 <;;; {(2,3)} = C x D, but A~C. (d) False since, by (b), the implication +- is false. . (e) [BB] True. Let x E A. Then x E AU B, so x E An B and, in particular, x E B. Thus, A <;;; B. Similarly, we have B <;;; A, so A = B. 29.
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