document.write( "Question 1209282: 1. (~(~Z v H) ⊃ ~T)
\n" ); document.write( "2. ((S ⊃ Z) ⊃ ~H)
\n" ); document.write( "∴ (Z ⊃ ~T)\r
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\n" ); document.write( "\n" ); document.write( "what's the next steps for this?
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Algebra.Com's Answer #848098 by math_tutor2020(3817) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Here is one way to do a derivation using a conditional proof.
\n" ); document.write( "I'll use arrow symbols in place of horseshoe symbols.
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Number		Statement	Line(s) Used	Reason
1		~(~Z v H) --> ~T
2		(S --> Z) --> ~H
:.		Z --> ~T
	3	Z		Assumption for Conditional Proof
	4	(Z & ~H) --> ~T	1	De Morgan’s Law
	5	Z --> (~H --> ~T)	4	Exportation
	6	~H --> ~T	5,3	Modus Ponens
	7	(S --> Z) --> ~T	2,6	Hypothetical Syllogism
	8	(~S v Z) --> ~T	7	Material Implication
	9	Z v ~S	3	Addition
	10	~S v Z	9	Commutation
	11	~T	8,10	Modus Ponens
12		Z --> ~T	3 - 11	Conditional Proof

\n" ); document.write( "I started with assuming Z is the case (line 3). Then I used the logic rules of inference and replacement to arrive at ~T (line 11)\r
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\n" ); document.write( "\n" ); document.write( "The assumption Z leading to ~T then allows us to prove Z --> ~T is a valid conclusion. \r
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\n" ); document.write( "\n" ); document.write( "Here's a way to do it using a direct proof.
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Number	Statement	Line(s) Used	Reason
1	~(~Z v H) --> ~T
2	(S --> Z) --> ~H
:.	Z --> ~T
3	(~S v Z) --> ~H	2	Material Implication
4	~(~S v Z) v ~H	3	Material Implication
5	(S & ~Z) v ~H	4	De Morgan’s Law
6	(S v ~H) & (~Z v ~H)	5	Distribution
7	(~Z v ~H) & (S v ~H)	6	Commutation
8	~Z v ~H	7	Simplification
9	Z --> ~H	8	Material Implication
10	(Z & ~H) --> ~T	1	De Morgan’s Law
11	(~H & Z) --> ~T	10	Commutation
12	~H --> (Z --> ~T)	11	Exportation
13	Z --> (Z --> ~T)	9, 12	Hypothetical Syllogism
14	(Z & Z) --> ~T	13	Exportation
15	Z --> ~T	14	Tautology

\n" ); document.write( "There might be a much more efficient pathway, but I'm not able to think of it right now.\r
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\n" ); document.write( "\n" ); document.write( "Caution: Tutor Edwin makes the mistake of using the Addition Rule on part of an expression rather than the entire thing.
\n" ); document.write( "It is NOT valid to go from ~(T ⊃ ~Z) to ~(T ⊃ (~Z v H))
\n" ); document.write( "Notice in this rule set the \"addition\" property is in the \"rules of inference\" sub-block. The rules of inference must apply to the entire line. In contrast a rule of replacement can apply to a portion of a line.
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