1.a. Write briefly about laws of logic and inference rules? Obtain the principle conjunctive normal form of (P ∧ Q)V(~PVQVR) ? 1.b. Obtain the principle disjunctive normal form of (P→ Q) ∧ (Q ↔R) ? Show that ( ∃ x)(P(x) ∧ Q(x)) ⇒ ( ∃ x)P(x) ∧ ( ∃ x)Q(x) . 2.a. Explain the term tautology? Show that [(p → q) → r] → [( p→q)→(p→r)] is tautology ? 2.b. Prove that the transitive closure R + of a relation R on set A is the smallest transitive relation on A containing R. 3.a. Define warshall ’ s algorithm? What is the negation of given statement? “ Ram will take job in a software company or higher studies ” 3.b. Explain in detail about types of functions with an example? If A={1,2,3,4}, B={a,b,c,d} Determine if the following functions are one to one or one to many by giving matrix representations i) f = {(1,a),(2,a),(3,b),(4,d)} ii) g = {(1,d),(2,b),(3,a),(4,a)}