Question 1003091
<table border=1><col width="300"><tr><th>Original Statement</th><th>Translation</th></tr><tr><td>If Alex does not score well on his final exams and Alex does not earn an academic scholarship, the he does not go to college.</td><td>(~F * ~E) -&gt; ~C</td></tr><tr><td>If Alex does not study, then he does not score well on his final exams.</td><td>~S -&gt; ~F</td></tr><tr><td>if Alex gets involved in too many extracurricular activities, then Alex does not study.</td><td>X -&gt; ~S</td></tr><tr><td>Alex gets involved in too many extracurricular activities.</td><td>X</td></tr><tr><td>Alex goes to college.</td><td>C</td></tr></table>


We have these premises
(~F * ~E) -> ~C
~S -> ~F
X -> ~S
X
C



The goal is to prove E. One way to do it is to use a derivation table and follow the rules of inference/rules of replacement
<table border=1><tr><th>Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td>(~F * ~E) -&gt; ~C</td><td></td><td></td></tr><tr><td>2</td><td>~S -&gt; ~F</td><td></td><td></td></tr><tr><td>3</td><td>X -&gt; ~S</td><td></td><td></td></tr><tr><td>4</td><td>X</td><td></td><td></td></tr><tr><td>5</td><td>C</td><td></td><td></td></tr><tr><td>:.</td><td>E</td><td></td><td></td></tr><tr><td>6</td><td>~S</td><td>3,4</td><td>MP</td></tr><tr><td>7</td><td>~F</td><td>2,6</td><td>MP</td></tr><tr><td>8</td><td>~~C</td><td>5</td><td>DN</td></tr><tr><td>9</td><td>~(~F * ~E)</td><td>1,8</td><td>MT</td></tr><tr><td>10</td><td>~~F v ~~E</td><td>9</td><td>DM</td></tr><tr><td>11</td><td>F v E</td><td>10</td><td>DN</td></tr><tr><td>12</td><td>E</td><td>11,7</td><td>DS</td></tr></table>


Abbreviations used:

MP = Modus Ponens
MT = Modus Tollens
DN = Double Negation
DM = De Morgan's Law
DS = Disjunctive Syllogism