solve step by step

PROBLEM 1 Are the following statements tautologies or not? (1p/part) a) (PVQ) V ( -PA-Q). b) (P -+ Q) v (Q - P). C) ((P -+ Q) A(Q -+ R)) - (P - R). d) ((PAQ) VR) + ((PAR)V(QAR)). PROBLEM 2 Let | A| = n. Find a formula (in terms of n) for the number of functions f : A + P(A) such that VI E A : x E f(x). PROBLEM 3 Consider the relation _ on Z given by x z y if there exist m E N and an odd integer k such that kx = 2" y. a) Prove that _ is an order relation on Z. (2p) b) Draw the Hasse diagram of