SOLUTION: please help me by showing the proof of the following using the rules of replacements..This propositions are logically equivalences and I need to see the proofs..thank you..
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-> SOLUTION: please help me by showing the proof of the following using the rules of replacements..This propositions are logically equivalences and I need to see the proofs..thank you..
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Question 1046573: please help me by showing the proof of the following using the rules of replacements..This propositions are logically equivalences and I need to see the proofs..thank you..
1. (P v Q)=>R = (P=>R)^(Q=>R)
2. P v (P^Q) = P
3. P ^ (PvQ) = P Answer by robertb(5830) (Show Source):
You can put this solution on YOUR website! 1. (P v Q)=>R => (P=>R)^(Q=>R)
Proof:
(P v Q)=>R
---> ~(P v Q) V R (Material implication)
---> (~P ^ ~Q) v R (de Morgan's)
---> (~P v R) ^ (~Q v R) (distributivity)
---> (P=>Q) ^ (Q=>R) (Material implication)
To prove (P=>R)^(Q=>R) => (P v Q)=>R, just reverse the steps in the preceding argument.
2. By addition, it is easy to see that P => P v (P^Q) .
To prove P v (P^Q) => P:
P v (P^Q)
---> (PvP)^(PvQ) (distributivity)
---> P^(PvQ) (idempotency)
---> P (simplification)
Therefore, P v (P^Q) = P.
