Discrete Mathematics with Graph Theory (3rd Edition) 167

Discrete Mathematics with Graph Theory (3rd Edition) 167 -...

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Section 6.1 165 (c) Here we want IE E9 AI = IE U AI - IE n AI = 11 - 6 = 5. (d) This asks for the number of elements in ECnN = (EUA)c. This is IUI-IEUAI = 15-11 = 4. 2. Suppose n people were surveyed. Let E be the set of those who had been emotionally abused and P the set of those who had been physically abused. We are given that lEI = .98n and IPI = .89n. We have n ;::: IE U PI = lEI + IPI - IE n PI = 1.87n - IE n PI and so IE n PI ;::: .87n. Thus, at least 87% of those surveyed had experienced both types of abuse. 3. Let U, J, F be the sets of all students in the class, those knowing JAVA and those knowing HTML, respectively. (a) IJ U HI = IJI + IHI -IJ n HI = 15 + 12 - 5 = 22. (b) IJ" HI = IJI-IJ n HI = 15 - 5 = 10. (c) IH" JI = IHI - IH n JI = 12 - 5 = 7. (d) IH E9 JI = IH U JI - IH n JI = 22 - 5 = 17. (e) IU" (J U H)I = IUI-IJ U HI = 30 - 22 = 8. 4. [BB] Let A be the set of people with undergraduate degrees in arts, S those with undergraduate degrees in science, and G
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Unformatted text preview: (a) This part asks for IA U S U GI. IAuSuGI IAI + lSI + IGI - IA n SI - IA n GI - IS n GI + IA n S n GI 200 + 95 + 120 -25 -55 - 40 + 5 = 300 (b) This part asks for IS " (A U G)I = lSI - IS n (A U GI. Now S n (A U G) = (S n A) U (S n G), so Isn (AU G)I IS n AI + IS n GI-I(S n A) n (S n G)I IS n AI + IS n GI - IS nAn GI = 25 + 40 -5 = 60. Therefore, IS" (A U GI = 95 -60 = 35. 5. Let P, G, and 0 denote the sets of children who purchased popsicles, gum, and candy bars, respec-tively. The number of children present was at least IP U G U 01 = IPI + IGI + 101 - IP n GI - IP n 01 - IG n 01 + IP n G n 01 = 10 + 7 + 12 -5 -6 -2 + IP n G n 01 = 16 + IP n G n 01 ;::: 16. Since there were only 15 children in all, the numbers don't tally. 6. Let E be the set of students taking English and M the set of students taking Math. (a) We want IE n MI. Now I(E U M)CI = 12, so IE U MI = 82 -12 = 70. Since IE U MI = lEI + IMI - IE n MI, IE n MI = lEI + IMI - IE U MI = 59 + 46 -70 = 35....
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