document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179696>Question 1179696</A>: <I><SPAN STYLE=\"word-break: break-all;\"> INSTRUCTIONS: Use natural deduction to derive the conclusion in each problem.\r<BR>\n" );
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document.write( "Prove this using natural deduction.<BR>\n" );
document.write( "NOTE: Use * for dot, v for wedge, ~ for tilde, = for triple bar (or copy and paste ≡), and > for horseshoe (or copy and paste ⊃ )\r<BR>\n" );
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document.write( "1.	N  ≡  F	 <BR>\n" );
document.write( "2.	~F v ~N	 <BR>\n" );
document.write( "3.	D ⊃ N	~(F v D)\r<BR>\n" );
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document.write( "1.	(B ⊃ G) • (F ⊃ N)	 <BR>\n" );
document.write( "2.	~(G * N)	/ ~(B * F)\r<BR>\n" );
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document.write( "1.	(J • R) ⊃ H	 <BR>\n" );
document.write( "2.	(R ⊃ H) ⊃ M	 <BR>\n" );
document.write( "3.	~(P v ~J)	/ M • ~P\r<BR>\n" );
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document.write( "1.	(F • H) ⊃ N	 <BR>\n" );
document.write( "2	F v S	 <BR>\n" );
document.write( "3.	H	/ N v S\r<BR>\n" );
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document.write( "Please any guidance on these 4 questions I'd greatly appreciate it!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850183 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850183\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve these natural deduction proofs step-by-step:\r<BR>\n" );
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document.write( "1. Proof:\r<BR>\n" );
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document.write( "N ≡ F<BR>\n" );
document.write( "~F v ~N<BR>\n" );
document.write( "D ⊃ N / ~(F v D)<BR>\n" );
document.write( "Proof:\r<BR>\n" );
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document.write( "(N ⊃ F) * (F ⊃ N) (1, Equivalence)<BR>\n" );
document.write( "N ⊃ F (4, Simplification)<BR>\n" );
document.write( "F ⊃ N (4, Simplification)<BR>\n" );
document.write( "~N ⊃ ~F (5, Contraposition)<BR>\n" );
document.write( "~F ⊃ ~N (6, Contraposition)<BR>\n" );
document.write( "~N v ~F (7, Implication)<BR>\n" );
document.write( "~N (2, 9, Resolution)<BR>\n" );
document.write( "~D (3, 10, Modus Tollens)<BR>\n" );
document.write( "~F (8, 10, Modus Ponens)<BR>\n" );
document.write( "~F * ~D (11, 12, Conjunction)<BR>\n" );
document.write( "~(F v D) (13, De Morgan's)<BR>\n" );
document.write( "Therefore, ~(F v D) is proven.\r<BR>\n" );
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document.write( "2. Proof:\r<BR>\n" );
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document.write( "(B ⊃ G) * (F ⊃ N)<BR>\n" );
document.write( "~(G * N) / ~(B * F)<BR>\n" );
document.write( "Proof:\r<BR>\n" );
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document.write( "B ⊃ G (1, Simplification)<BR>\n" );
document.write( "F ⊃ N (1, Simplification)<BR>\n" );
document.write( "~G v ~N (2, De Morgan's)<BR>\n" );
document.write( "~B v G (3, Implication)<BR>\n" );
document.write( "~F v N (4, Implication)<BR>\n" );
document.write( "~B v ~N (5, 6, Resolution)<BR>\n" );
document.write( "~F v ~G (5, 7, Resolution)<BR>\n" );
document.write( "~B (8, 9, Resolution)<BR>\n" );
document.write( "~F (8, 10, Resolution)<BR>\n" );
document.write( "~B * ~F (10, 11, Conjunction)<BR>\n" );
document.write( "~(B * F) (12, De Morgan's)<BR>\n" );
document.write( "Therefore, ~(B * F) is proven.\r<BR>\n" );
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document.write( "3. Proof:\r<BR>\n" );
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document.write( "(J * R) ⊃ H<BR>\n" );
document.write( "(R ⊃ H) ⊃ M<BR>\n" );
document.write( "~(P v ~J) / M * ~P<BR>\n" );
document.write( "Proof:\r<BR>\n" );
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document.write( "~P * ~~J (3, De Morgan's)<BR>\n" );
document.write( "~P (4, Simplification)<BR>\n" );
document.write( "~~J (4, Simplification)<BR>\n" );
document.write( "J (6, Double Negation)<BR>\n" );
document.write( "~(J * ~R) (1, Implication)<BR>\n" );
document.write( "~J v ~R (8, De Morgan's)<BR>\n" );
document.write( "~R (7, 9, Disjunctive Syllogism)<BR>\n" );
document.write( "R ⊃ H (1, Exportation)<BR>\n" );
document.write( "M (2, 11, Modus Ponens)<BR>\n" );
document.write( "M * ~P (5, 12, Conjunction)<BR>\n" );
document.write( "Therefore, M * ~P is proven.\r<BR>\n" );
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document.write( "4. Proof:\r<BR>\n" );
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document.write( "(F * H) ⊃ N<BR>\n" );
document.write( "F v S<BR>\n" );
document.write( "H / N v S<BR>\n" );
document.write( "Proof:\r<BR>\n" );
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document.write( "~F v N (1, Exportation)<BR>\n" );
document.write( "F (2, Assumption)<BR>\n" );
document.write( "N (4, 5, Disjunctive Syllogism)<BR>\n" );
document.write( "N v S (6, Addition)<BR>\n" );
document.write( "S (2, Assumption)<BR>\n" );
document.write( "N v S (8, Addition)<BR>\n" );
document.write( "H (3, Copy)<BR>\n" );
document.write( "F * H (5, 10, Conjunction)<BR>\n" );
document.write( "N (1, 11, Modus Ponens)<BR>\n" );
document.write( "N v S (12, Addition)<BR>\n" );
document.write( "N v S (5-7, 8-9, 13, Conditional Proof)<BR>\n" );
document.write( "Therefore, N v S is proven. </SPAN>\n" );
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