Question 1206424
<font color=black size=3>
Here are the <a href="https://www.algebra.com/algebra/homework/Conjunction/logic-rules-of-inference-and-replacement.lesson">Rules of Inference</a> that you should have on a reference sheet or memorized.


I'll do an indirect proof. This is also known as a proof by contradiction.
The idea is to assume the opposite of the conclusion. Then show that assumption leads to a contradiction; which therefore must mean the original conclusion is indeed the case.


I'll use the ampersand & in place of the dot.
I'll use the arrow -> in place of the horseshoe. 
<table border = "1" cellpadding = "5"><tr><td colspan=2>Number</td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td></td><td>(K v L) -> (M & N)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>(N v O) -> (P & ~K)</td><td></td><td></td></tr><tr><td>:.</td><td></td><td>~K</td><td></td><td></td></tr><tr><td></td><td>3</td><td>~(~K)</td><td></td><td>Assumption for Indirect Proof</td></tr><tr><td></td><td>4</td><td>K</td><td>3</td><td>Double Negation</td></tr><tr><td></td><td>5</td><td>K v L</td><td>4</td><td>Addition</td></tr><tr><td></td><td>6</td><td>M & N</td><td>1,5</td><td>Modus Ponens</td></tr><tr><td></td><td>7</td><td>N</td><td>6</td><td>Simplification</td></tr><tr><td></td><td>8</td><td>N v O</td><td>7</td><td>Addition</td></tr><tr><td></td><td>9</td><td>P & ~K</td><td>2,8</td><td>Modus Ponens</td></tr><tr><td></td><td>10</td><td>~K</td><td>9</td><td>Simplification</td></tr><tr><td></td><td>11</td><td>K & ~K</td><td>4,10</td><td>Conjunction</td></tr><tr><td>12</td><td></td><td>~K</td><td>3 - 11</td><td>Indirect Proof</td></tr></table>
The original conclusion is ~K


Line 3 is where we assume the opposite of that conclusion.
Following the logic of lines 3 to 11, we arrive at K & ~K which is a contradiction. So that allows us to conclude ~K at the end.
</font>