# 5 c5k k1 10 5 pts suppose that a and b are sets prove

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5 C(5,k) k=1 10. (5 pts.) Suppose that A and B are sets. Prove the following using the definition of the terms subset and intersection . If A = A B, then A B.

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MAD2104/Final Exam Page 5 of 8 11. (5 pts.) What is the minimum number of students required in your Discrete Mathematics class to be sure that at least 6 have birthdays occurring in the same month this year? Explain. 12. (10 pts.) Suppose A = { ,3,4} and B = { ,3,{ }}. Then A B = A × B = |P(A)| = A - B = A B =
MAD2104/Final Exam Page 6 of 8 13. (5 pts.) Suppose that R is an equivalence relation on a nonempty set A. Recall that for each a ε A, the equivalence class of a is the set [a] = {s | (a,s) ε R}. Prove the following proposition: If [a] [b] ≠ ∅ , then [a] = [b]. Hint: The issue is the set equality, [a] = [b], under the hypothesis that [a] [b] ≠ ∅ . So pretend [a] [b] ≠ ∅ and use this to show s ε [a] s ε [b], and s ε [b] s ε [a]. Be explicit regarding your use of the relational properties of R. 14. (5 pts.) Let {a n } be defined by the formula a n = 4n + 2 for n = 1,2,3, .... The sequence {b n } is defined recursively by b 1 = 6 and b n+1 = b n + 4 for n = 1,2,3, .... Give a proof by induction that a n = b n for n = 1,2,3, ....

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MAD2104/Final Exam Page 7 of 8 15. (5 pts.) Construct the ordered rooted binary tree representing the following expression: (C - (A B)) = ((C - A) (C - B)) 16. (5 pts.) Suppose G 1 = (V 1 ,E 1 )is an undirected graph with adjacency matrix 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 , and G 2 = (V 2 ,E 2 )is an undirected graph with adjacency matrix 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 . Are G 1 and G 2 isomorphic?? Either display an explicit graph isomorphism g:V 1 V 2 or give an invariant that one graph has but the other doesn’t have. [Warning: The graphs are obviously not simple. Drawing them might help.]
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