Question 1029291
Problem #15


<table border=1 cellpadding=3><tr><th>Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td>(~D v E) &amp; (~D v ~F)</td><td></td><td></td></tr><tr><td>2</td><td>(E &amp; ~F) -&gt; ~G</td><td></td><td></td></tr><tr><td>3</td><td>~D -&gt; ~G</td><td></td><td></td></tr><tr><td>:.</td><td>~G</td><td></td><td></td></tr><tr><td>4</td><td>~D v E</td><td>1</td><td>Simp</td></tr><tr><td>5</td><td>~D v ~F</td><td>1</td><td>Simp</td></tr><tr><td>6</td><td>D -&gt; E</td><td>4</td><td>MI</td></tr><tr><td>7</td><td>D -&gt; ~F</td><td>5</td><td>MI</td></tr><tr><td>8</td><td>E -&gt; (~F -&gt; ~G)</td><td>2</td><td>Exp</td></tr><tr><td>9</td><td>D -&gt; (~F -&gt; ~G)</td><td>6,8</td><td>HS</td></tr><tr><td>10</td><td>(D &amp; ~F) -&gt; ~G</td><td>9</td><td>Exp</td></tr><tr><td>11</td><td>(~F &amp; D) -&gt; ~G</td><td>10</td><td>Comm</td></tr><tr><td>12</td><td>~F -&gt; (D -&gt; ~G)</td><td>11</td><td>Exp</td></tr><tr><td>13</td><td>D - &gt; (D -&gt; ~G)</td><td>7,12</td><td>HS</td></tr><tr><td>14</td><td>(D &amp; D) -&gt; ~G</td><td>13</td><td>Exp</td></tr><tr><td>15</td><td>D -&gt; ~G</td><td>14</td><td>Taut</td></tr><tr><td>16</td><td>G -&gt; D</td><td>3</td><td>Trans</td></tr><tr><td>17</td><td>G -&gt; ~G</td><td>16,15</td><td>HS</td></tr><tr><td>18</td><td>~G v ~G</td><td>17</td><td>MI</td></tr><tr><td>19</td><td>~G</td><td>18</td><td>Taut</td></tr></table>


Abbreviations Used:


Comm: Commutation
Exp: Exportation
HS: Hypothetical Syllogism
MI: Material Implication
Simp: Simplification
Taut: Tautology
Trans: Transposition


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Problem #24


<table border=1 cellpadding=3><tr><th>Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td>~O</td><td></td><td></td></tr><tr><td>:.</td><td>~Q -&gt; ~(O &amp; P)</td><td></td><td></td></tr><tr><td>2</td><td>~O v Q</td><td>1</td><td>Add</td></tr><tr><td>3</td><td>Q v ~O</td><td>2</td><td>Comm</td></tr><tr><td>4</td><td>(Q v ~O) v ~P</td><td>3</td><td>Add</td></tr><tr><td>5</td><td>~(~Q &amp; O) v ~P</td><td>4</td><td>DM</td></tr><tr><td>6</td><td>(~Q &amp; O) -&gt; ~P</td><td>5</td><td>MI</td></tr><tr><td>7</td><td>~Q -&gt; (O -&gt; ~P)</td><td>6</td><td>Exp</td></tr><tr><td>8</td><td>~Q -&gt; (~O v ~P)</td><td>7</td><td>MI</td></tr><tr><td>9</td><td>~Q -&gt; ~(O &amp; P)</td><td>8</td><td>DM</td></tr></table>


Abbreviations Used:

Add: Addition
Comm: Commutation
DM: De Morgan's Law
Exp: Exportation
MI: Material Implication