Question 255427: (R->B)&(B->~C),D <-> (P&~K),R&C |- M
I recently started taking an intensive Logic class and I am falling behind. Can you please help me solve this exercise so i can get an idea how to do these type of exercises? thanks so much
JT
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
1. (R -> B) & (B -> ~C)
2. D <-> (P & ~K)
3. R & C /:. M
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4. (B -> ~C) & (R -> B) 1 Commutation
5. R -> B 1 Simplification
6. B -> ~C 4 Simplification
7. C & R 3 Commutation
8. R 3 Simplification
9. C 7 Simplification
10. B 5,8 Modus Ponens
11. ~C 6,10 Modus Ponens
12. C & ~C 9,11 Conjunction
13. (C & ~C) v M 12 Addition
14. F v M 13 Contradiction
15. M 14 See note below
Note: Recall that P v Q is true when either P or Q (or both) are true. If P is known to be false, then the truth value of P v Q is entirely dependent on Q. In other words, if P is known to be false, then P v Q is true if Q is true or P v Q is false if Q is false. If you aren't convinced of this observation, make a truth table.
Also, remember that P & Q is only true when both P and Q are true. Since P can only be true or false (but not both), then ~P will be the opposite making P & ~P automatically false for both truth values of P. Again, make a truth table if you aren't convinced.
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