Let W = ((a,b,c, d)| 5a - b = 0) be a subspace of R. Then a basis for W is O {(1,5,0,0)} O {(1,5,0,0),(0,0,0,1)} O {(1,5,0,0),(0,0,1,0),(0,0,0,1)} O {(5,1,0,0),(0,0,1,0),(0,0,0,1)}

The correct statements are: Let R = {(1,2), (2,3), (3,4)}. {(1, 2), (2, 3), (3, 4), (1, 3), (1, 4), (2, 4), (4,4)} is the transitive closure of R. Let R = {(1,2), (2,3), (3,4)}. {(1, 2), (2, 3), (3, 4), (1, 1), (2, 2), (3, 3), (4,4)} is the reflexive closure of R. Let R = {(1,2), (2,3), (3,4)}. {(1, 2), (2, 3), (3, 4), (2, 1), (3, 2), (4, 3), (4, 4)) is the symmetric closure of R.

Find a countermodel for each of the following wffs: Jx p(x) → Vx p(x) Vx (p(x, f(x)) → p(x, y))

Let i, j ∈ Za and w, x ∈ Zb with i, j different from each other and w, x different from each other. When a function f : Za → Zb is chosen at random, what is Pr[f(j) = x | f(i) = w]?

For each function, indicate the class Θ(g(n)) the function belongs to. Prove each assertion. 1. f(n) = 2n ln(n+2)2 + (n+2)2 ln(n/2) 2. f(n) = 2n+1 + 3n+1 3. f(n) = floor(ln(n))

Given interpretations (a) (Vx) (y) P(x, y) → (x)(Vy) P(x, y) (b) (Vx) [P(x) → Q(x)] → [(x) P(x) → (Vx)Q(x)] (c) (Vx)[A(x)]' → [(Vx)A(x)]' to prove that each of the following wffs is not valid.

Is it true that for any nfa M = (Q,E,8.90,F) the complement of L(M)is equal to the set {w e E*: 8* (40,w) n. F= Ø}? If so, prove it. If not, give a counterexample. Is it true that for every nfa M = (Q, £,8,40.F)the complement of L (M)is equal to the set {w e £* : 8*; (g0,w) e (Q – F) = Ø}? If so, prove it; if not, give a counterexample.

Consider F = {BC -> D, B -> E, CE -> D, E -> CA, BF -> G} and R(A,B,C,D,E,F,G) 1.Find a minimal cover for F. 2.Find all many keys for R.

Let f (f(n) and g(n)) be asymptotically nonnegative functions. Using the basic definition of Θ notation, prove that max(f(n), g(n)) = Θ(f(n) + g(n)),

Exercise 12.9 Given a weighted (connected) undirected graph G = (V, E, w), with m =n+ 0(1) edges (i.e., the number of edges is at most the number of vertices plus some constant), give an O(n)-time algorithm to compute the minimum spanning tree of G. Prove the correctness and the running time of your algorithm. (Hint: Think of how one can eliminate an edge from being in the MST.)