document.write( "Question 1209177: Premise:
\n" ); document.write( "H & ( C & T )
\n" ); document.write( "~ ( ~ F & T )
\n" ); document.write( "Conclusion:
\n" ); document.write( "F
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Algebra.Com's Answer #847767 by math_tutor2020(3816) $\"\"$ $\"About$

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\n" ); document.write( "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. \r
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\n" ); document.write( "\n" ); document.write( "Here's a direct derivation
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Number	Statement	Line(s) Used	Reason
1	H & (C & T)
2	~( ~F & T )
:.	F
3	(H & C) & T	1	Association
4	T & (H & C)	3	Commutation
5	T	4	Simplification
6	~(~T)	5	Double Negation
7	~(~F) v ~T	2	De Morgan’s Law
8	F v ~T	7	Double Negation
9	~T v F	8	Commutation
10	F	9, 6	Disjunctive Syllogism

\n" ); document.write( "Here's a list of rules of inference and replacement\r
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\n" ); document.write( "\n" ); document.write( "Another way to do the derivation is to use an indirect proof (aka proof by contradiction)
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Number	Statement	Line(s) Used	Reason
1		H & (C & T)
2		~( ~F & T )
:.		F
	3	~F		Assumption for Indirect Proof
	4	~(~F) v ~T	2	De Morgan’s Law
	5	F v ~T	4	Double Negation
	6	~T	5, 3	Disjunctive Syllogism
	7	(H & C) & T	1	Association
	8	T & (H & C)	7	Commutation
	9	T	8	Simplification
	10	T & (~T)	9, 6	Conjunction
11		F	3 - 10	Indirect Proof

\n" ); document.write( "Line 3 is where we assume the opposite of the conclusion we want to arrive at.
\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "The assumption ~F led to a contradiction, which means the opposite (F) must be a valid conclusion.
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