1 1 1 m r s m r m r 0 1 1 1 0 1 c construct the

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1 1 1 M R S = M R M R = 0 1 1 1 0 1 (c) Construct the matrix representing the relation T = R -1 that is the inverse of R. 1 0 0 M T = M R T = 1 1 0 1 1 1 _________________________________________________________________ 4. (10 pts.) (a) What is the numerical value of the postfix expression below? 3 2 * 2 5 3 - 8 4 / * - = 6 2 5 3 - 8 4 / * - = 36 5 3 - 8 4 / * - = 36 2 8 4 / * - = 36 2 2 * - = 36 4 - = 32 (b) What is the numerical value of the prefix expression below? + - * 2 3 5 / 2 3 4 = + - * 2 3 5 / 8 4 = + - * 2 3 5 2 = + - 6 5 2 = + 1 2 = 3
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TEST3/MAD2104 Page 3 of 4 _________________________________________________________________ 5. (15 pts.) (a) Draw a directed graph G 1 whose adjacency matrix is given on the left below.// V 1 = { a, b, c, d }. 0 1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 . (b) Now draw the underlying undirected graph G 2 for the directed graph G 1 of part (a) of this problem.// V 2 = { a, b, c, d }. (c) Is G 2 = (V 2 ,E 2 ), above, isomorphic to the simple graph G 3 = (V 3 ,E 3 ) given below? Either display an isomorphism f:V 2 V 3 or very briefly explain why there is no such function by revealing an invariant that one graph has that the other doesn’t. Yes, define f:V 2 V 3 by f(a) = b, f(b) = c, f(c) = a, and f(d) = d. Then { a ,b } ε E 2 { f(a), f(b) } = { b, c } ε E 3 , { b ,c } ε E 2 { f(b), f(c) } = { c, a } ε E 3 , { b ,d } ε E 2 { f(b), f(d) } = { c, d } ε E 3 , { c ,d } ε E 2 { f(c), f(d) } = { a, d } ε E 3 . [You could also use f(a) = b, f(b) = c, f(c) = d, and f(d) = a.] _________________________________________________________________ 6. (10 pts.) Recall that the composition of two relations R and S on a set A is given by S R = { (a,c) ε A × A ( b)(b ε A and (a,b) ε R and (b,c) ε S)}. Also, recall that the n th composition power of a relation on a set is defined recursively by R 1 = R, and for each n ε , if n 1, then R n+1 = R n R.
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