Mind your P's and Q's. Please use only the provided theorems to solve the problem and use justification throughout. We are trying to conclude exists x R(x)

5. Mind Your P's and Q's (18 points) Using the logical inference rules and equivalences we have given, write a formal proof that given Vu (By P(x,y) -Q(x). Va (-R(2) (QC) V-P(2, 2)), and 3.c P(1,2), you can conclude that 3.c R(2). Identity Domination PAT=p pVF=p PVT=T PAF=F Idempotency pvp =p pp=p Commutativity pVq=qVp PAquap Associativity (p Va) Vr=pv (Vr) ( pq) Ar=pA (Ar) Distributivity pA (Vr) = ( pq) (par) pv (Ar) = (pVg) (p Vr) Absorption Abcoration Negation pv (19) =p pA (PVq) =p PV-p=T PA-p=F Double Negation DeMorgan's Laws -(pVq) = npA9 -(19) = PV- p=p Contrapositive Law of Implication p+q=pVq p +95 + P 5. Mind Your P's and Q's (18 points) Using the logical inference rules and equivalences we have given, write a formal proof that given Vu (By P(x,y) -Q(x). Va (-R(2) (QC) V-P(2, 2)), and 3.c P(1,2), you can conclude that 3.c R(2). Identity Domination PAT=p pVF=p PVT=T PAF=F Idempotency pvp =p pp=p Commutativity pVq=qVp PAquap Associativity (p Va) Vr=pv (Vr) ( pq) Ar=pA (Ar) Distributivity pA (Vr) = ( pq) (par) pv (Ar) = (pVg) (p Vr) Absorption Abcoration Negation pv (19) =p pA (PVq) =p PV-p=T PA-p=F Double Negation DeMorgan's Laws -(pVq) = npA9 -(19) = PV- p=p Contrapositive Law of Implication p+q=pVq p +95 + P