Do A and C

Do C, D, E, H

Do A

Below are several proofs showing that two logical expressions are logically equivalent Label the steps in each proof with the law used to obtain each proposition from the previous proposition. The first line in the proof does not have a label (-) A V p) (pvc) (p) (-p) ( ap) 9V (pap) qvp-p) QUE (PV) - (DA) --PV) v (pad) -DA-Q) (p (PA-) V (PA) Dagva Solution (-ova) -( 0) (pval ( pa) Conditional identity (-A-a) (pad) De Morgan's law (PA-a) v ( p a) Double negation law DA(-ava) Distributive PAT Domplement law Identity (C) Irv ( -p) TV (-ryp) rv (rvp) (rvr)vp Use the laws of propositional logic to prove the following (a) -p-+-99- Solution -P -9 -- - Conditional identity PV- Double negation law -gv Commutative law 9 p Conditional identity (b) p^(p-a)p Solution A PA(-p ) PA(-- Va). Conditional identity PA(Va) Double negation law A bsorption law P ( p a) A(p-) =p - (AP) -P-(4-1)=4-(Vr) (p-1)(a ) = (PA) -(pv (-p ))-PA- PAA-) V (PA-A-) PA- P -pv (PA) (A-1) -- (a) Show that the two sentences below are logically equivalent Express each pair of sentences using a logical expression her prove whether the two expressions are logically equivalent Note you can assume that x and y are real numbes 30 60 irrational, thenx is rational and i s not rational, then x is an irrational number * Ifx is a rational number and y is an irrational number then xy is as irrational number - Ifx is a rational number and xey is a rational number then y is a rational number