4. Consider the formula A = Vx (p(x) →q(x)) → (Vxp(x) →Vaq(x)). (a) Show that A is valid. (b) Show that the converse of A is not valid.

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.

Justify (argue, prove, ...) that the following formula is valid Vx A(x) V Vx B(x) → Vx (A(x) V B(x))

Problem 1. Use the Boolean algebra derivation (instead of truth table) to prove or answer the questions. p↔q = (p→q)^ (q→→p) pvq-(p→q) →q (p+q)' =p' q Is ((p = q)) ((pr) ➡(qr)) a tautology, why? Provide the proof or disproof. Is (p q) (p →q) a tautology, why? Provide the proof or disproof.

prove the following using an equational style proof if x does not occur free in A

a) Show that: (Vx)(A → (B = C)) → ((\x)(A → B) = (Vx)(A → → C)) b) Show that: (\x)((A v B) → C) → (Vx) (A → C) c) [Show that: (\x)(A → (B ^ C)) → (\x)(A → B) d) Prove (AV (Vx) BC = (\x)B(AV C), as long as x not free in A. (Hint: Translate first to Standard notation)

Solve the following:

Please write a formal solution. Use math, do not use words only.