Discrete Mathematics with Graph Theory (3rd Edition) 170

Discrete Mathematics with Graph Theory (3rd Edition) 170 -...

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168 Solutions to Exercises (b) The number of elements which belong to Al and A2 but to no other Ai is IA 1 nA2 ,(A3UA4)1. We determine this number and then multiply by 6 because there are six ways for elements to belong to exactly two of the sets. Since (AI n A2) n (A3 U A4) = n n A3) U n n A4), I(A 1 n A2) n U A4)1 = IAI n n A31 + IAI n n A41-IAl n n A3 n n n A41 = 5 + 5 -1 = 9. Thus I(A 1 n A2) , U A4)1 = IAI n A21 -I(AI n A2) n U A4)1 = 12 - 9 = 3. There are 6 x 3 = 18 elements in exactly two of the four subsets. 11. (a) [BB] Let A and B be the set of integers between 1 and 500 which are divisible by 3 and 5, respectively. The question asks for IA UBI. This number is IAI + IBI - n BI = l5~0 J + l5g0 J - l5 1 0 5 0 J = 166 + 100 - 33 = 233. (b) [BB] Let A and B be as in (a) and G be the set of integers between 1 and 500 which are divisible by 6. The question asks for A, (BUG)I and so the number we want is IAI-IAn (BuG)I. Now An (B U G) = (A n B) U n G), so the number of elements here is l5 1 0 5 0 J + l5~0 J -l5 3 0 0 0 J = 33 + 83 - 16 = 100. Finally, we obtain , (B U G)I = 166 - 100 = 66. 12. (a) Since 500 = 225 3 , an integer is relatively prime to 500 if and only if it is not divisible by 2 or by
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