document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204702>Question 1204702</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use indirect proof (IP) together with the eight rules of implication and ten rules of replacement to prove that they are valid. Be sure to include the justification for each line, and offset lines as appropriate for indirect proof.\r<BR>\n" );
document.write( "\n" );
document.write( "1. C ⊃ (N • I)<BR>\n" );
document.write( "2. (N v P) ⊃ (I ⊃ ~C)		/~C<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841142 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=841142\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The conclusion is ~C<BR>\n" );
document.write( "With indirect proofs, or proofs by contradiction, we assume the opposite of the conclusion is the case. Then we show some contradiction arises from this assumption. Thereby proving the original conclusion to be the case.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Number</TD><TD ></TD><TD >Statement</TD><TD >Line(s) Used</TD><TD >Reason</TD></tr><tr><TD >1</TD><TD ></TD><TD >C --> (N & I)</TD><TD ></TD><TD ></TD></tr><tr><TD >2</TD><TD ></TD><TD >(N v P) --> (I --> ~C)</TD><TD ></TD><TD ></TD></tr><tr><TD >:.</TD><TD ></TD><TD >~C</TD><TD ></TD><TD ></TD></tr><tr><TD ></TD><TD >3</TD><TD >~(~C)</TD><TD ></TD><TD >Assumption for Indirect Proof</TD></tr><tr><TD ></TD><TD >4</TD><TD >C</TD><TD >3</TD><TD >Double Negation</TD></tr><tr><TD ></TD><TD >5</TD><TD >N & I</TD><TD >1, 4</TD><TD >Modus Ponens</TD></tr><tr><TD ></TD><TD >6</TD><TD >I & N</TD><TD >5</TD><TD >Commutation</TD></tr><tr><TD ></TD><TD >7</TD><TD >N</TD><TD >5</TD><TD >Simplification</TD></tr><tr><TD ></TD><TD >8</TD><TD >I</TD><TD >6</TD><TD >Simplification</TD></tr><tr><TD ></TD><TD >9</TD><TD >N v P</TD><TD >7</TD><TD >Addition</TD></tr><tr><TD ></TD><TD >10</TD><TD >I --> ~C</TD><TD >2, 9</TD><TD >Modus Ponens</TD></tr><tr><TD ></TD><TD >11</TD><TD >~C</TD><TD >10, 8</TD><TD >Modus Ponens</TD></tr><tr><TD ></TD><TD >12</TD><TD >C & (~C)</TD><TD >4, 11</TD><TD >Conjunction</TD></tr><tr><TD >13</TD><TD ></TD><TD >~C</TD><TD >3 - 12</TD><TD >Indirect Proof</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "More info:<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Conjunction/logic-rules-of-inference-and-replacement.lesson\">Logic Rules of Inference and Replacement</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A similar logic problem<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Proofs/Proofs.faq.question.1204701.html\">https://www.algebra.com/algebra/homework/Proofs/Proofs.faq.question.1204701.html</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );