No need solutions, just final answer will do

What is simplest form of the logical expression (y=0)=(x>1)? A. (y=0)(x>1) B. (y0v(x>1) D. (y>0) (x>1) Select one: a. D O b. A OC B O d. C. Is the following statement a tautology, contingent or contradiction? (P=(Q^ R)) =((~PvQ) ^ ( PR)) Select one: O a. a contingent O b. a contradiction O c. a tautology Consider the relation R on Z given by: R= {(x, y): (x, y e Z)^ (x y)} Which of the following proof/proofs is/are correct to show that R is symmetric? Proof 1: Let a, b e Z, (a, b) e R, a + b. This implies that b#a. Hence, (b, a) ER Therefore, R is symmetric. Proof 2: V x,y e Z, (x, y) = R, X#y. This implies that y # X. Hence, (y,x) ER Therefore, R is symmetric. Proof 3: V x, y e N, (x, y) e R,x=y. This implies that y # X. Hence, (y, x) ER Therefore, R is symmetric. Proof 4: 2,3 Z, (2, 3) e R, 2 + 3. This implies that 3 + 2. Hence, (3, 2) ER Therefore, R is symmetric Select one: O a. Only Proof 2 is correct. O b. All the proofs are correct. Oc Only Proof 1 and Proof 2 are correct. O d. Except Proof 4, all are correct. Oe. None of them are correct. Let U be the Universal set. Consider the relation R on P(U) (the power set of U) given by: R= {(A, B): (A, B e P(U))^(A CB)} Which of the following statement is correct? Select one: O a. Ris symmetric, but not reflexive and not transitive. Ob. Ris reflexive, symmetric, and transitive. O c. R is not reflexive, not symmetric, and not transitive. O d. R is transitive, but not reflexive and not symmetric. What is the truth value of the predicate (Vx R, (x>1= x->x))? Select one: O a. False O b. True O c. Unknown