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# E the total number of ways to assign truth values to

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(e) The total number of ways to assign truth values to five true-false problems by using the letters T and F is given by the sum below. 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 .I fA=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.) SupposeA={ ,3,4} andB={ ,3,{ }}. Then A B= A ×B= |P(A)| = A-B= A
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 =4 n+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 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 1110 1001 0110 , and G 2 V 2 ,E 2 )is an undirected graph with adjacency matrix 0111 . 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
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e The total number of ways to assign truth values to five...

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