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.. 1. (P

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.

3. I leave this up to you.

RELATED QUESTIONS
can you help me solve this proof using the first 13 rules of inference please?... (answered by jim_thompson5910)

Please help me answer this one natural proof CORRECTLY using the 8 rules of implication... (answered by robertb)

solve the system of equations using the addition method. equ 1: 2x-1/5y=-5 equ 2:... (answered by ewatrrr)

Please help me create this proof using the 18 rules of inference. I have been working on... (answered by jim_thompson5910)

Solve the following proof using natural deduction (rules of replacement and rules of... (answered by robertb)

II. Write the following propositions in symbols with the first propositions as p and the (answered by Solver92311)

Please help me by showing me how to work out this equation using the Square Root Property (answered by Alan3354)

I need to find integer replacements for the following: 2< [x] < 5 Please show... (answered by checkley71)

Please show me the rule to solve the equ.... (answered by ewatrrr)