propositional logic

Question

A)  ((P->Q)->Q)  ==>  P V Q is given

How i solved

((~P V Q)-->Q) 

~(~P V Q) V Q

(P and ~Q) V Q

(P V Q) and ( ~Q VQ)

(P V Q) and Q

after this.. how to proceed?

B)  ( P <--> Q ) ==> (P and Q ) V ( ~p and ~Q)..............how ??

Comments

At 4^th step of A     ( ~Q V Q) = True

Answer 1

( P <--> Q )   <--> (P->Q) and (Q->P)

<--> (~P V Q) and (~Q V P)

<--> (~P  and (~Q V P)) V ( Q and (~Q V P))

<--> ((~P  and ~Q) V (~P and P)) V ((Q  and ~Q) V (Q and P))

<--> ((~P  and ~Q) V False) V (False V (Q and P))

<--> (~P  and ~Q) V (Q and P)

==> (~P  and ~Q) V (Q and P)

Answer 2

$(A)$$((P \rightarrow Q) \rightarrow Q) \\ \equiv ((\sim P \vee Q) \rightarrow Q) \\ \equiv (\sim(\sim P \vee Q) \vee Q) \\ \equiv ((P \wedge \sim Q) \vee Q) \\ \equiv ((P \wedge \sim Q) \vee Q) \\ \equiv (P \vee Q) \wedge (Q \vee \sim Q)) \\ \equiv (P \vee Q) \wedge T \\ \equiv (P \vee Q)$

$(B)$$(P \leftrightarrow Q) \\ \equiv (P \rightarrow Q) \wedge(Q \rightarrow P) \\ \equiv (\sim P \vee Q) \wedge(\sim Q \vee P) \\ \equiv (\sim P \wedge \sim Q) \vee (P \wedge \sim P) \vee (Q \wedge \sim Q)\vee (Q \wedge P) \\ \equiv (\sim P \wedge \sim Q) \vee F \vee F \vee (Q \wedge P) \\ \equiv (P \wedge Q) (\sim P \wedge \sim Q)$