Question: 7. Construct proofs for the following more challenging problems. a. ( ~ AV ~ B) - ~ C, (A ) F), (~ F = (D

7. Construct proofs for the following more challenging problems. a. ( ~ AV ~ B) - ~ C, (A ) F), (~ F = (D . ~ E)), ~ (D ) H), (EPH) I.. ~ C b. ~(AV (B ) T)), (A . C) v (WD ~ D), ~ (P VT) - D, ~ P = ~ ( T . S ) / . . ~ W * C. ~ ( F . G) V ~ ( H . K), ( B . X ) = ~ K, ~ (X V Y), ~ ( H . F) DY 1. . ~(G . A) *d. (A . F) _ (C v G), ~ (Cv ( F . G)), F = ~ (X . Y), ~ (W -X) /.. ~(AV X) e. ~ (B . (H V J)), (AV C) = (BV D), ~ (A D D), ( B . F) DJ I . . ~ F f. ~ P = ~ (Q ) R), ~ (PV (S DT)), (Z D W), ~(RV T) - ~(S . W) 1.. ~Zmake a great many proofs a great deal eas- ier and will also allow us to carry out proofs that are not possible with only the basic rules and replacement rules. They will thus round out our system of sen- tential logic. SUMMARY OF REPLACEMENT RULES DOUBLE NEGATION (D.N.) DUPLICATION (DUP.) P :: ~ ~P P :: (PVP) P :: (P .P) 168 Unit 8 Replacement Rules COMMUTATION (COMM.) ASSOCIATION (ASSOC.) (p v q) :: (q v p) ( (pv q) vr) :: (pv (qvr)) (P . q) :: (9. P) ( ( p . q) . r) :: (p. (q.r)) CONTRAPOSITION (CONTRAP.) DE MORGAN'S (DEM.) (p P q) :: (~ q) ~p) ~ ( p v q) :: ( ~ p. ~ q) ~ ( P . q) :: ( ~ pv ~ q) BICONDITIONAL EXCHANGE (B.E.) CONDITIONAL EXCHANGE (C.E.) (p = q) :: ((p > q) . (q > p)) ( p > q) :: (~pv q) DISTRIBUTION (DIST.) EXPORTATION (EXP.) (p . (q vr)) :: ((p . q) v (p.r)) ( (p . q) > r) :: (p >(q>r)) (pv ( q . r)) : ((pv q). (pvr))as puzzles, as intellectual challenges; proofs are rather like mazes, where you have to reach the conclusion by working through to the right path. If you take it in this spirit, the process of learning to construct proofs should prove to be intellectually rewarding. SUMMARY OF RULES OF INFERENCE MODUS PONENS (M.P.) MODUS TOLLENS (M.T.) HYPOTHETICAL SYLLOGISM (H.S.) p P q p. > q p q p ~ 9 9 5 r 1 .. q 1 .. ~ P 1.. Par SIMPLIFICATION (SIMP.) CONJUNCTION (CONJ.) DILEMMA (DIL.) P . q p q p >q 1 .. P 1 .. q r Ds 1 . . P . q pvr 1 .. q vs DISJUNCTIVE SYLLOGISM (D.S.) ADDITION (ADD.) pvq pvq P ~ p 1 .. pvq 1 . . pvq 1 :.q 1 . . P DEFINITIONS 1. A constant is a term that has a definite, particular value

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