Question 1209177
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I'll assume that these particular F and T symbols do not represent "false" and "true", but rather just any logical statement. Your professor should have chosen different letters. 


Here's a direct derivation
<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td>H & (C & T)</td><td></td><td></td></tr><tr><td>2</td><td>~( ~F & T )</td><td></td><td></td></tr><tr><td>:.</td><td>F</td><td></td><td></td></tr><tr><td>3</td><td>(H & C) & T</td><td>1</td><td>Association</td></tr><tr><td>4</td><td>T & (H & C)</td><td>3</td><td>Commutation</td></tr><tr><td>5</td><td>T</td><td>4</td><td>Simplification</td></tr><tr><td>6</td><td>~(~T)</td><td>5</td><td>Double Negation</td></tr><tr><td>7</td><td>~(~F) v ~T</td><td>2</td><td>De Morgan’s Law</td></tr><tr><td>8</td><td>F v ~T</td><td>7</td><td>Double Negation</td></tr><tr><td>9</td><td>~T v F</td><td>8</td><td>Commutation</td></tr><tr><td>10</td><td>F</td><td>9, 6</td><td>Disjunctive Syllogism</td></tr></table>
Here's a list of <a href="https://www.algebra.com/algebra/homework/Conjunction/logic-rules-of-inference-and-replacement.lesson">rules of inference and replacement</a>


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Another way to do the derivation is to use an indirect proof (aka proof by contradiction)
<table border = "1" cellpadding = "5"><tr><td colspan="2">Number</td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td></td><td>H & (C & T)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>~( ~F & T )</td><td></td><td></td></tr><tr><td>:.</td><td></td><td>F</td><td></td><td></td></tr><tr><td></td><td>3</td><td>~F</td><td></td><td>Assumption for Indirect Proof</td></tr><tr><td></td><td>4</td><td>~(~F) v ~T</td><td>2</td><td>De Morgan’s Law</td></tr><tr><td></td><td>5</td><td>F v ~T</td><td>4</td><td>Double Negation</td></tr><tr><td></td><td>6</td><td>~T</td><td>5, 3</td><td>Disjunctive Syllogism</td></tr><tr><td></td><td>7</td><td>(H & C) & T</td><td>1</td><td>Association</td></tr><tr><td></td><td>8</td><td>T & (H & C)</td><td>7</td><td>Commutation</td></tr><tr><td></td><td>9</td><td>T</td><td>8</td><td>Simplification</td></tr><tr><td></td><td>10</td><td>T & (~T)</td><td>9, 6</td><td>Conjunction</td></tr><tr><td>11</td><td></td><td>F</td><td>3 - 10</td><td>Indirect Proof</td></tr></table>
Line 3 is where we assume the opposite of the conclusion we want to arrive at.
From there a chain event of dominoes fall over to lead to T & (~T) which is a contradiction. One of T or ~T is false, while the other is true. This contradiction means our assumption must be the opposite.


The assumption ~F led to a contradiction, which means the opposite (F) must be a valid conclusion.
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