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

Mathematical logic Is (((P→Q→P)→P) a tautology? Please be as clear as possible. Explain your answer. Thank you very much.

Please explain, as simplistically (but thoroughly) as possible, how to develop a k-map from a truth table to be used to simplify an equation.Examples: xz + (xy + ~z) ~xyz + yz + x~y

Show that the compound statement is a tautology.((p ∨ q) ∧ ￢p) → q

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)

A Boolean function F (A,B,C) of truth table at below（PLS step by step） 1，Provide the minterm expansion of F .  2，Provide the minterm expansion and the maxterm expansion of F’s complement, respectively. Assume that we don’t care the value of F when ABC=001. i. What is the minterm expansion of F?  ii. Choose a proper value for the don’t care term and provide the shortest possible expression of POS (product of sum) for F . (Hint: Minimize the number of maxterms)

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