Discrete Mathematics with Graph Theory (3rd Edition) 169

Discrete Mathematics with Graph Theory (3rd Edition) 169 -...

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Section 6.1 167 and so 890 = 750 + 400 + 100 + 50 - 328 - 25 - 12 - 35 - 8 - 33 + 17 + 4 + 9 + 2 -10 nAn G n 01· Thus, 10 01 = 891- 890 = 1. (b) We want IG" (0 u A u e)l, and we know Ie" (0 u A u e)1 = IGI-IG n (0 u A u e)1 = 100 -IG n (0 u A u e)l· Now G n (0 u A u e) = (G n 0) u (G n A) u (G n e) and so IGn (0 uAue)1 = IGn 01 + IGnAI + IGnel-IGn 0 nAI-IGnOnel -IG n B n + IG nOn A n = 25 + 35 + 33 - 17 - 9 - 2 + 1 = 66 So the number we want is 100 - 66 = 34. (c) There are six possibilities to consider. The number who bought orange juice and apple juice but no other kind of juice is 1(0 n A) " (G u e)1 = 10 n AI-I(O n n (G u e)1 = 10 n AI-I(O nAn G) u (0 nAn e)1 = 10nAI-IOnAnGI-IOnAnCi + 10nAnGnei = 328 - 17 - 4 + 1 = 308. Similarly, the number who bought orange juice and grapefruit juice but no other kind of juice is 25-17 -9+1 = O. The number who bought only orange juice and citrus punch is 12-4-9+1 = O. The number who bought only apple juice and grapefruit juice is 35 - 17 - 2 + 1 = 17. The number who bought only apple juice and citrus punch is 8 - 4 - 2 + 1 = 3. The number who bought only grapefruit juice and citrus punch is 33 - 2 - 9 + 1 = 23. Thus, we conclude that
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