document.write( "Question 1171807: 1. (J v F) v M
\n" ); document.write( " 2. (J v M) ⊃ ~ P
\n" ); document.write( " 3. ~F/~(F v P)
\n" ); document.write( "4. M Assumption for Indirect Proof
\n" ); document.write( "5.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "12. ~(F v P) \r
\n" ); document.write( "\n" ); document.write( "I think there are 12 lines to get to the conclusion?
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #796719 by math_tutor2020(3817) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Here's one way to do the derivation
\n" ); document.write( "I'm using an arrow symbol in place of the horseshoe, or sideways \"U\", symbol.
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

Number		Statement	Lines Used	Reason
1		(J v F) v M
2		(J v M) -> ~P
3		~F
:.		~(F v P)
	4	~~(F v P)		Assumption for Indirect Proof
	5	F v P	4	Double Negation
	6	P v F	5	Commutation
	7	~~P v F	6	Double Negation
	8	~P -> F	7	Material Implication
	9	(J v M) -> F	2,8	Hypothetical Syllogism
	10	J v (F v M)	1	Association
	11	J v (M v F)	10	Commutation
	12	(J v M) v F	11	Association
	13	~~(J v M) v F	12	Double Negation
	14	~(J v M) -> F	13	Material Implication
	15	~F -> ~~(J v M)	14	Transposition
	16	~F -> (J v M)	15	Double Negation
	17	~F -> F	16,9	Hypothetical Syllogism
	18	F	17,3	Modus Ponens
	19	~F & F	3,18	Conjunction
20		~(F v P)	4-19	Indirect Proof

\n" ); document.write( "The idea is to assume the opposite of the conclusion. So we assume the opposite of ~(F v P), which is ~~(F v P) or simply F v P.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Then we use the rules of inference to show that a contradiction happens because of this. The contradiction occurs in line 19 when we say that ~F & F.
\n" ); document.write( "For example, we could say that F = \"an object can fly\", meaning that ~F = \"an object cannot fly\". The statement ~F & F means \"the object cannot fly AND the object can fly\". This is one example showing why we get a contradiction.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Since we get a contradiction, we then can conclude the opposite of the assumption is the case. So it's the opposite of ~~(F v P), which is ~(F v P) and that concludes the proof.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It's probably possible to have the proof done in 12 lines. I used more lines to be more thorough with the step by step process. As I finished up the table, I realized that I probably could have taken a more efficient route.
\n" ); document.write( " \n" ); document.write( "

\n" );