Question 1157891
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The conclusion is ~K
The opposite of this is ~~K, or more simply just K
The idea is to assume ~~K or K is the case and show how this leads to a contradiction. Note how lines 4 and 12 contradict each other, so this is the key to this indirect proof. The term "indirect proof" is the same as "proof by contradiction".


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<tr><td colspan = 2>Number</td><td>Statement</td><td>Lines 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 colspan=2>Conclusion</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 & M</td><td>6</td><td>Commutation</td></tr>
<tr><td></td><td>8</td><td>N</td><td>7</td><td>Simplification</td></tr>
<tr><td></td><td>9</td><td>N v O</td><td>8</td><td>Addition</td></tr>
<tr><td></td><td>10</td><td>P & ~K</td><td>2,9</td><td>Modus Ponens</td></tr>
<tr><td></td><td>11</td><td>~K & P</td><td>10</td><td>Commutation</td></tr>
<tr><td></td><td>12</td><td>~K</td><td>11</td><td>Simplification</td></tr>
<tr><td></td><td>13</td><td>K & ~K</td><td>4,12</td><td>Conjunction</td></tr>
<tr><td>14</td><td></td><td>~K</td><td>3-13</td><td>Indirect Proof</td></tr>
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