Question 1005719: How do I prove:
P, ~(P&Q) ⊢ ~Q
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Step 1) Use De Morgan's Law to go from ~(P&Q) to ~P v ~Q
Step 2) Use the disjunctive syllogism property with ~P v ~Q and P (which is really ~~P) to get ~Q
Step 3) If you assume the complete opposite of the conclusion, we will assume ~~Q is true. That's the same as saying Q is true
Step 4) From step 3, we assume Q is true. But step 2) says that ~Q is true. They both can't be true at the same time. If Q is true, then ~Q is false or vice versa. This is a contradiction. So that invalidates the assumption made in step 3. So the complete opposite must be true. Otherwise we get a contradiction.
In short, if you assume Q to be true, then we end up with a contradiction. So ~Q must be true. This wraps up the proof.
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