show that (P->(q v r)) is logically equivalent with ((p ^~q)->r) Solution We wan
ID: 3651532 • Letter: S
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show that (P->(q v r)) is logically equivalent with ((p ^~q)->r)Explanation / Answer
We want to show that (P v Q) > R is equivalent to (P > R) & (Q > R). To prove that two statements are equivalent is to prove that each entails the other, so I'd first show that (P > R) & (Q > R) follows from (P v Q) > R, then show that (P v Q) > R follows from (P > R) & (Q > R). 1. (P v Q) > R Premise 2. ~(P v Q) v R 1 Material Implication 3. (~P & ~Q) v R 2 De Morgan's Law 4. R v (~P & ~Q) 3 Commutation 5. (R v ~P) & (R v ~Q) 4 Distribution 6. (~P v R) & (R v ~Q) 5 Commutation 7. (~P v R) & (~Q v R) 6 Commutation 8. (P > R) & (~Q v R) 7 Material Implication 9. (P > R) & (Q > R) 8 Material Implication Thus, (P v Q) > R entails (P > R) & (Q > R).
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