Question 1009929
Natural Deduction



<table border=1><tr><th>Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td>M -&gt; (R ^ E)</td><td></td><td></td></tr><tr><td>2</td><td>(E v H) -&gt; G</td><td></td><td></td></tr><tr><td>:.</td><td>M -&gt; G</td><td></td><td></td></tr><tr><td>3</td><td>~(E v H) v G</td><td>2</td><td>MI</td></tr><tr><td>4</td><td>(~E ^ ~H) v G</td><td>3</td><td>DM</td></tr><tr><td>5</td><td>G v (~E ^ ~H)</td><td>4</td><td>Comm</td></tr><tr><td>6</td><td>(G v ~E) ^ (G v ~H)</td><td>5</td><td>Dist</td></tr><tr><td>7</td><td>G v ~E</td><td>6</td><td>Simp</td></tr><tr><td>8</td><td>~E v G</td><td>7</td><td>Comm</td></tr><tr><td>9</td><td>E -&gt; G</td><td>8</td><td>MI</td></tr><tr><td>10</td><td>~M v (R ^ E)</td><td>1</td><td>MI</td></tr><tr><td>11</td><td>(~M v R) ^ (~M v E)</td><td>10</td><td>Dist</td></tr><tr><td>12</td><td>(~M v E) ^ (~M v R)</td><td>11</td><td>Comm</td></tr><tr><td>13</td><td>~M v E</td><td>21</td><td>Simp</td></tr><tr><td>14</td><td>M -&gt; E</td><td>13</td><td>MI</td></tr><tr><td>15</td><td>M -&gt; G</td><td>14,9</td><td>HS</td></tr></table>




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Conditional Proof (alternative method)


Assume M is true. Show this leads to G being true.


<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>M -&gt; (R ^ E)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>(E v H) -&gt; G</td><td></td><td></td></tr><tr><td>:.</td><td></td><td>M -&gt; G</td><td></td><td></td></tr><tr><td></td><td>3</td><td>M</td><td></td><td>ACP</td></tr><tr><td></td><td>4</td><td>R ^ E</td><td>1,3</td><td>MP</td></tr><tr><td></td><td>5</td><td>E ^ R</td><td>4</td><td>Comm</td></tr><tr><td></td><td>6</td><td>E</td><td>5</td><td>Simp</td></tr><tr><td></td><td>7</td><td>E v H</td><td>6</td><td>Add</td></tr><tr><td></td><td>8</td><td>G</td><td>2,7</td><td>MP</td></tr><tr><td>9</td><td></td><td>M -&gt; G</td><td>3-8</td><td>CP</td></tr></table>



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Abbreviations/Acronyms Used (applies to either method)


ACP = Assumption for Conditional Proof
Add = Addition
Comm = Commutation
CP = Conditional Proof
Dist = Distribution
DM = De Morgan's Law
HS = Hypothetical Syllogism
MI = Material Implication
MP = Modus Ponens
Simp = Simplification