document.write( "Question 1005719: How do I prove: \r
\n" ); document.write( "\n" ); document.write( "P, ~(P&Q) ⊢ ~Q\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #621989 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Step 1) Use De Morgan's Law to go from ~(P&Q) to ~P v ~Q\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Step 2) Use the disjunctive syllogism property with ~P v ~Q and P (which is really ~~P) to get ~Q\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "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\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "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.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "

\n" );