document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190422>Question 1190422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> MAT 145: Topics In Contemporary Math\r<BR>\n" );
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document.write( "QUESTION 7\r<BR>\n" );
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document.write( "Determine if the argument is valid or invalid using a truth table.\r<BR>\n" );
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document.write( "Premise: p rightwards arrow tilde q\r<BR>\n" );
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document.write( "Premise: p logical or q\r<BR>\n" );
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document.write( "Conclusion: p </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #822065 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=822065\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "You can type a tilde for \"not\" or \"negation\". Look above the TAB key\r\n" );
document.write( "to the left of the 1 key.  You have to hold the shift key down. ~\r\n" );
document.write( "You can make a right arrow for conditional\" or \"if then\" with a dash \r\n" );
document.write( "- and a greater than > to get ->.  Use a little \"v\" for \"logical or\".\r\n" );
document.write( "For \"logical and\", you can use \"&\".</PRE><BR>\n" );
document.write( " Determine if the argument is valid or invalid using a truth table.\r<BR>\n" );
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document.write( "Premise:    p->~q<BR>\n" );
document.write( "Premise:    pvq<BR>\n" );
document.write( "Conclusion: p<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( "Put & between the premises and -> before the conclusion\r\n" );
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document.write( "[(p->~q)&(pvq)]->p\r\n" );
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document.write( "You have to use parentheses and brackets so that there are only two\r\n" );
document.write( "things inside a parentheses or bracket with a v, &, or -> between \r\n" );
document.write( "them. \r\n" );
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document.write( "Start with a column for each single letter\r\n" );
document.write( "for a statement on the left, and build it \r\n" );
document.write( "up piece by piece.\r\n" );
document.write( "Each heading must have either \r\n" );
document.write( "p\r\n" );
document.write( "q\r\n" );
document.write( "~p\r\n" );
document.write( "~q\r\n" );
document.write( "( ) v ( )\r\n" );
document.write( "( ) & ( )\r\n" );
document.write( "( ) -> ( )\r\n" );
document.write( "Single letters or letters with a tilde before them can \r\n" );
document.write( "replace the ( ), and the ( ) can have ~ before them.\r\n" );
document.write( "You can use brackets [ ] to enclose ( ). But remember\r\n" );
document.write( "each parentheses and each brackets can at most only \r\n" );
document.write( "contain two things with v, &, or -> between them. \r\n" );
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document.write( "Start with this:\r\n" );
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document.write( "| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |\r\n" );
document.write( "|   |   |    |       |     |               ||                    |   \r\n" );
document.write( "|   |   |    |       |     |               ||                    |     \r\n" );
document.write( "|   |   |    |       |     |               ||                    | \r\n" );
document.write( "|   |   |    |       |     |               ||                    | \r\n" );
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document.write( "Put TTFF in the p column and TFTF in the q column:\r\n" );
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document.write( "| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |\r\n" );
document.write( "| T | T |    |       |     |               ||                    |   \r\n" );
document.write( "| T | F |    |       |     |               ||                    |     \r\n" );
document.write( "| F | T |    |       |     |               ||                    | \r\n" );
document.write( "| F | T |    |       |     |               ||                    |\r\n" );
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document.write( "Next we have a ~q column.  We put the opposite of what was in the q\r\n" );
document.write( "column:\r\n" );
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document.write( "| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |\r\n" );
document.write( "| T | T |  F |       |     |               ||                    |   \r\n" );
document.write( "| T | F |  T |       |     |               ||                    |     \r\n" );
document.write( "| F | T |  F |       |     |               ||                    | \r\n" );
document.write( "| F | T |  F |       |     |               ||                    |\r\n" );
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document.write( "Next we have a p->~q column.  The rule for -> is \r\n" );
document.write( "\"put T for everything except T->F, where there's a T on the \r\n" );
document.write( "left of -> and an F on the right of ->.  That's the only time you \r\n" );
document.write( "put F.\" You have to look at the previous columns to see what's on \r\n" );
document.write( "the left of -> and what's on the right of it.\r\n" );
document.write( "\r\n" );
document.write( "| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |\r\n" );
document.write( "| T | T |  F |   F   |     |               ||                    |   \r\n" );
document.write( "| T | F |  T |   T   |     |               ||                    |     \r\n" );
document.write( "| F | T |  F |   T   |     |               ||                    | \r\n" );
document.write( "| F | F |  T |   T   |     |               ||                    |\r\n" );
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document.write( "Next we have a pvq column.  The rule for v is \r\n" );
document.write( "\"put T for everything except FvF, where there's an F on both\r\n" );
document.write( "sides of the v. That's the only time you put F.\" You have to \r\n" );
document.write( "look at the previous columns to see what's on the left of v \r\n" );
document.write( "and what's on the right of it.\r\n" );
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document.write( "| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |\r\n" );
document.write( "| T | T |  F |   F   |  T  |               ||                    |   \r\n" );
document.write( "| T | F |  T |   T   |  T  |               ||                    |     \r\n" );
document.write( "| F | T |  F |   T   |  T  |               ||                    | \r\n" );
document.write( "| F | F |  T |   T   |  F  |               ||                    |\r\n" );
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document.write( "Next we have a (p->~q)&(pvq) column. The rule for & is \r\n" );
document.write( "\"put F for everything except T&T, where there's an T on both\r\n" );
document.write( "sides of the &. That's the only time you put T.\" You have to \r\n" );
document.write( "look at the previous columns to see what's on the left of & \r\n" );
document.write( "and what's on the right of it.\r\n" );
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document.write( "| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |\r\n" );
document.write( "| T | T |  F |   F   |  T  |        F      ||                    |   \r\n" );
document.write( "| T | F |  T |   T   |  T  |        T      ||                    |     \r\n" );
document.write( "| F | T |  F |   T   |  T  |        T      ||                    | \r\n" );
document.write( "| F | F |  T |   T   |  F  |        F      ||                    |\r\n" );
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document.write( "Finally we have a [(p->~q)&(pvq)]->p column.  We already had a ->\r\n" );
document.write( "column. The rule, remember, for -> is \"put T for everything except \r\n" );
document.write( "T->F, where there's a T on the left of -> and an F on the right of\r\n" );
document.write( "->.  That's the only time you put F.\" You have to look at the \r\n" );
document.write( "previous columns to see what's on the left of -> and what's on the \r\n" );
document.write( "right of it.\r\n" );
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document.write( "| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |\r\n" );
document.write( "| T | T |  F |   F   |  T  |        F      ||                 T  |   \r\n" );
document.write( "| T | F |  T |   T   |  T  |        T      ||                 T  |     \r\n" );
document.write( "| F | T |  F |   T   |  T  |        T      ||                 F  | \r\n" );
document.write( "| F | F |  T |   T   |  F  |        F      ||                 T  |\r\n" );
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document.write( "Oh, oh! there's a F in that last column, so the argument is invalid.\r\n" );
document.write( "There must be only T's in the last column for an argument to be\r\n" );
document.write( "valid.\r\n" );
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document.write( "Be sure to learn those rules for ~,v,->, and &.  There is only one \r\n" );
document.write( "other, and that's equivalence <->.  The rule for that is to put T if\r\n" );
document.write( "what are on both sides are the same (both T's or both F's), and F if \r\n" );
document.write( "they aren't the same.\r\n" );
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document.write( "Edwin</PRE>  </SPAN>\n" );
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