document.write( "Question 1193421: Prove the following using indirect proof:\r
\n" ); document.write( "\n" ); document.write( "1. (A ∨ B) ⊃ C
\n" ); document.write( "2. (∼A ∨ D) ⊃ E / C ∨ E\r
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Algebra.Com's Answer #825440 by math_tutor2020(3817) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Here's how I'd do the derivation table
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Number		Statement	Line(s) Used	Reason
1		(A v B) -> C
2		(~A v D) -> E
	:.	C v E
	3	~(C v E)		Assumption for Indirect Proof
	4	~C & ~E	3	De Morgan’s Law
	5	~C	4	Simplification
	6	~E	4	Simplification
	7	~(A v B)	1,5	Modus Tollens
	8	~A & ~B	7	De Morgan’s Law
	9	~A	8	Simplification
	10	~(~A v D)	2,6	Modus Tollens
	11	~~A & ~D	10	De Morgan’s Law
	12	A & ~D	11	Double Negation
	13	A	12	Simplification
	14	A & ~A	13,9	Conjunction
15		C v E	3-14	Indirect Proof, aka proof by contradiction

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\n" ); document.write( "\n" ); document.write( "Further explanation:
\n" ); document.write( "In place of the horseshoe symbols, I used arrows.\r
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\n" ); document.write( "\n" ); document.write( "The idea is to negate the conclusion C v E to get ~(C v E) as the starting point of the indirect proof or proof by contradiction (this is done on line 3).
\n" ); document.write( "The goal is to try to find two statements from that starting point that clash with one another.
\n" ); document.write( "That contradiction will then lead back to C v E being the only possibility.\r
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\n" ); document.write( "\n" ); document.write( "On lines 9 and 13, I got ~A and A respectively. One or the other must happen, but both cannot be simultaneously the case.
\n" ); document.write( "It's like saying a light switch is off and on at the same time.
\n" ); document.write( "This is the key contradiction needed to conclude that ~(C v E) cannot be the case. Therefore, C v E is a valid conclusion.\r
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\n" ); document.write( "\n" ); document.write( "Notice how the \"number\" column has been indented for the subcase of assuming the ~(C v E). It's like going off on a separate hypothetical branch. Once we realize a contradiction happens with that branch, we then snap back to the opposite of ~(C v E) in other words C v E
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