document.write( "Question 1193814: Prove using Indirect Proof\r
\n" ); document.write( "\n" ); document.write( "1. B ⊃ (K • M)
\n" ); document.write( "2. (B • M) ⊃ (P ≡ ∼P) / ∼B \n" ); document.write( "

Algebra.Com's Answer #825875 by math_tutor2020(3816) $\"\"$ $\"About$

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\n" ); document.write( "This is one way to do the derivation using an indirect proof (aka proof by contradiction).
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Number		Statement	Line(s) Used	Reason
1		B -> (K & M)
2		(B & M) -> (P = ~P)
:.		~B
	3	B		Assumption for Indirect Proof
	4	K & M	1,3	Modus Ponens
	5	M	4	Simplification
	6	B & M	3, 5	Conjunction
	7	P = ~P	2, 6	Modus Ponens
	8	(P -> ~P) & (~P -> P)	7	Material Equivalence
	9	P -> ~P	8	Simplification
	10	~P v ~P	9	Material Implication
	11	~P	10	Taulogy
	12	~P -> P	8	Simplification
	13	~~P v P	12	Material Implication
	14	P v P	13	Double Negation
	15	P	14	Taulogy
	16	P & ~P	15, 11	Conjunction
17		~B	3 - 16	Indirect Proof

\n" ); document.write( "The conclusion we want to arrive at is ~B
\n" ); document.write( "Assume that the opposite is the case and we assume B (line 3)
\n" ); document.write( "The idea is to see if it generates a contradiction.\r
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\n" ); document.write( "\n" ); document.write( "The line P & ~P is a contradiction because one portion is true while the other is false, which makes P & ~P always false.
\n" ); document.write( "Or perhaps another approach is to look at line 7 where we have P = ~P. This isn't possible because if P is true then ~P is false, and vice versa. There's no way to have true equal false. So we could have stopped at that point to find the contradiction.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, the negation of the assumption must be the case and we have ~B instead as the proper conclusion. \r
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\n" ); document.write( "\n" ); document.write( "I used arrows in place of horseshoe symbols.
\n" ); document.write( "I used ampersands (&) in place of the dots.
\n" ); document.write( "Instead of a triple equal sign, I used a regular equal sign.
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