document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162984>Question 1162984</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is the following argument valid? Provide a proof for your answer.<BR>\n" );
document.write( "p =⇒ (q ∧ r)<BR>\n" );
document.write( "∼q<BR>\n" );
document.write( "− − − − − − − − − −<BR>\n" );
document.write( "∼p </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #786915 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=786915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The argument is valid. We can prove it as such using a proof by contradiction.\r<BR>\n" );
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document.write( "The conclusion is ~p, read as \"not p\". This is the opposite of statement p. Assume that p is the case.\r<BR>\n" );
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document.write( "If p is true, then so is q ^ r. This is because of the first line reading p -> (q ^ r). Use the modus ponens rule here. \r<BR>\n" );
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document.write( "q ^ r means that both q and r are true at the same time. Therefore, q is the case. But the second line says that ~q is also the case as well. \r<BR>\n" );
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document.write( "We have q and ~q clash to form the contradiction. \r<BR>\n" );
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document.write( "In short, we assume the opposite of the conclusion (we assume p) and it leads to a contradiction (q and ~q). So making the assumption that p is the case will not work.\r<BR>\n" );
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document.write( "Therefore, the opposite of the original assumption must be true and the conclusion of ~p is valid.\r<BR>\n" );
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document.write( "---------------------------------------------------------\r<BR>\n" );
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document.write( "Let's look at a truth table. This is where we list out all the possible truth values of p,q,r and build logical compound statements from them to form the argument as a whole\r<BR>\n" );
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document.write( "The first three columns are p,q,r which alternate with true (T) or false (F) values. This lists all the possible ways to have three T's or F's placed together.\r<BR>\n" );
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document.write( "The fourth column has us combine q and r to get q ^ r. This column is only true if both q and r are true together; otherwise it is false.\r<BR>\n" );
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document.write( "Then we construct a conditional p -> (q ^ r). This is only false if p is true while (q ^ r) is false.\r<BR>\n" );
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document.write( "Next up, form the column for ~q, which is the opposite of q. If q is true, then ~q is false, and vice versa.\r<BR>\n" );
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document.write( "Lastly, form the column for ~p, which is the conclusion. This will be the opposite of the p column.\r<BR>\n" );
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document.write( "Here's what the truth table looks like<BR>\n" );
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document.write( "<tr><TD >p</TD><TD >q</TD><TD >r</TD><TD >q ^ r</TD><TD >p -> (q ^ r)</TD><TD >~q</TD><TD >~p</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr>\n" );
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document.write( "and here's a truth table showing just the premises and the conclusion (I removed columns p, q, r, and  q ^ r)<BR>\n" );
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document.write( "<tr><TD >p -> (q ^ r)</TD><TD >~q</TD><TD >~p</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD></tr>\n" );
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document.write( "This is to help give the table a more simplified look.\r<BR>\n" );
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document.write( "We can see that there are no cases where we have all true premises lead to a false conclusion. An invalid argument is one where we have all true premises but a false conclusion. A valid argument is the opposite of that.\r<BR>\n" );
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document.write( "So this is another way to see we have a valid argument.\r<BR>\n" );
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