Question 1008772
Proof by contradiction


<table border=1><tr><th colspan="2">Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td></td><td>(R v S) -&gt; (H &amp; ~G)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>(K v R) -&gt; (G v ~H)</td><td></td><td></td></tr><tr><td>.:</td><td></td><td>~R</td><td></td><td></td></tr><tr><td></td><td>3</td><td>~~R</td><td></td><td>AIP</td></tr><tr><td></td><td>4</td><td>R</td><td>3</td><td>DN</td></tr><tr><td></td><td>5</td><td>R v S</td><td>4</td><td>Add</td></tr><tr><td></td><td>6</td><td>R v K</td><td>4</td><td>Add</td></tr><tr><td></td><td>7</td><td>K v R</td><td>6</td><td>Comm</td></tr><tr><td></td><td>8</td><td>H &amp; ~G</td><td>1,5</td><td>MP</td></tr><tr><td></td><td>9</td><td>~G &amp; H</td><td>8</td><td>Comm</td></tr><tr><td></td><td>10</td><td>H</td><td>8</td><td>Simp</td></tr><tr><td></td><td>11</td><td>~G</td><td>9</td><td>Simp</td></tr><tr><td></td><td>12</td><td>G v ~H</td><td>2,7</td><td>MP</td></tr><tr><td></td><td>13</td><td>~H</td><td>12,11</td><td>DS</td></tr><tr><td></td><td>14</td><td>H &amp; ~H</td><td>10,13</td><td>Conj</td></tr><tr><td>15</td><td></td><td>~(~~R)</td><td>3-14</td><td>IP</td></tr><tr><td>16</td><td></td><td>~R</td><td>15</td><td>DN</td></tr></table>


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Acronyms/Abbreviations Used


Add = Addition
AIP = Assumption for Indirect Proof
Comm = Commutation
Conj = Conjunction
DM = De Morgan's Law
DN = Double Negation
DS = Disjunctive Syllogism
IP = Indirect Proof (aka proof by contradiction)
MP = Modus Ponens
Simp = Simplification