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# USF MATH 300 - Formal Methods

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Formal MethodsKey to Homework Assignment 5February 28, 20075. Let A = {1, 2}. List the members of P(A).P(A) = {∅, {1}, {2}, {1,2}}.7. Is there a set with exactly 12 subsets? Explain.No. We saw in class that if A is a finite set, then |P(A)| = 2|A|. But 12 6= 2n, for anynonnegative integer n.8. A certain set A has exactly 15 proper subsets. How many me mbers does P(A) have?|P(A)| = 16. Any set S has exactly one nonproper subset, S itself. So the total numb erof subsets is 15 + 1 = 16.9. A certain set S has exactly 64 subsets. If a new element is added to S , how manysubsets will the new set have? Explain.Since 26= 64, |S| = 6. If we add one element to S, and call the new set T, then T has7 elements and hence |P(T )| = 27= 128.10. Determine which of the following statements are true.(a) ∅ ∈ {∅}. True.(b) ∅ ⊆ {∅}. True.(e) π/4 ∈ {{π/4}}. False. The set on the right-hand side has a single element, {π/4},and π/4 6= {π/4}.(f) {π/4} ⊆ {{π/4}}. False. In order for the set {π/4} to be a subset of {{π/4}},the sole element of {π/4}, π/4, must be an element of {{π/4}}. But we saw inpart (e) that this is not the case.(i) {π/4} ⊆ {π/4, {π/2}}. True.11. Determine which of the following statements are true1(e) Since ∅ is a member of {∅}, ∅ = {∅}. False. The hypothesis is true, but theconclusion is false. In order for ∅ = {∅} to be true, we require that ∅ ⊆ {∅}and {∅} ⊆ ∅. The first containment is true: the empty set is a subset of everyset. However, the second containment is false: {∅} contains one element, but ∅contains no elements.21. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 6}, B = {3, 4, 5, 6}, and C = {3, 6, 9}.List the members of each of the following sets.(a) A ∪ B = {1, 3, 4, 5, 6}.(e) (A ∪ B) ∩ C = {3, 6}(i) A0∪ B0= {2, 4, 5, 7, 8, 9, 10} ∪ {1, 2, 7, 8, 9, 10} = {1, 2, 4, 5, 7, 8, 9, 10}26. Draw Venn diagrams illustrating (A0)0. See me if you have any questions about theVenn diagrams.27. Draw Venn diagrams illustrating A ∪ (B ∩ C) and (A ∪ B) ∩ (A ∪ C).28. (a) Draw the Venn diagram associated with Theorem 2.6(c). (Theorem 2.6(c) saysthat (A ∩ B)0= A0∪ B0.)(b) Draw the Venn diagram associated with Theorem 2.6(d). (Theorem 2.6(c) saysthat A − B = A ∩

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