document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=253525>Question 253525</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My brain is 54 years old and it is trying to understand discrete structures.  My first hurdle is creating a truth table for ~(p^q) v (pvq). Maybe I should withdraw from my class??? :-)  I appreciate any assistance.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #185875 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20086)</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=185875\">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;\"><font size = 4 color = \"indigo\"><b>\r\n" );
document.write( "~(p^q) v (pvq).\r\n" );
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document.write( "Letters stand for sentences.  Some sentences are true and some are false.\r\n" );
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document.write( "Fro example let p = \"Today is Monday\" and q = \"This month is August\".\r\n" );
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document.write( "Sometimes p is true and q is true, say on a Monday in August.\r\n" );
document.write( "Sometimes p is true and q is false, say on a Monday in March.\r\n" );
document.write( "Sometimes p is false and q is true, say on a Tuesday in August.\r\n" );
document.write( "Sometimes p is false and q is false, say on a Tuesday in March.\r\n" );
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document.write( "~(p^q) v (pvq).\r\n" );
document.write( "That says:\r\n" );
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document.write( "It's not both Monday and August or it's Monday or August.\r\n" );
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document.write( "That's a compound sentence and it is hard to analyze, just reading it. \r\n" );
document.write( "That's why we need truth tables to sort out the various possibilities \r\n" );
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document.write( "There are only four possibilities to consider.\r\n" );
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document.write( "1.  p is true and q is true\r\n" );
document.write( "2.  q is true and q is false\r\n" );
document.write( "3.  p is false and q is true\r\n" );
document.write( "4   p is false and q is false\r\n" );
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document.write( "So we start with those four possibilities in a table:\r\n" );
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document.write( "The binary operators are ^ and v, \"and\" and \"or\". They always have letters\r\n" );
document.write( "on both sides of them.\r\n" );
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document.write( "The unary operator is ~, \"not\".  It can only have one letter on the right \r\n" );
document.write( "side of it.\r\n" );
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document.write( "^ means \"and\".  In order to be true it must have true statements on both\r\n" );
document.write( "sides of it.  It is false any other time.\r\n" );
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document.write( "v means \"or\".  In order for it to be true it only needs to have just one \r\n" );
document.write( "true statement on either side of it.  It is only false when it has false\r\n" );
document.write( "sentences on BOTH sides.\r\n" );
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document.write( "~ before a letter means that the sentence that follows it is false.\r\n" );
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document.write( "So the truth table for a statement that has two sentences, that is, \r\n" );
document.write( "letters, p and q, starts out as this:\r\n" );
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document.write( "        p  q\r\n" );
document.write( "case 1  T  T\r\n" );
document.write( "case 2  T  F\r\n" );
document.write( "case 3  F  T\r\n" );
document.write( "case 4  F  F\r\n" );
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document.write( "To build a truth table for ~(p^q) v (pvq)\r\n" );
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document.write( "we have to make headings for each of these: \r\n" );
document.write( "A.  p\r\n" );
document.write( "B.  q\r\n" );
document.write( "C.  (p^q)\r\n" );
document.write( "D.  ~(p^q) \r\n" );
document.write( "E.  (pvq)\r\n" );
document.write( "F.  ~(p^q) v (pvq)\r\n" );
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document.write( "We start with the four cases for letters p, q, steps A and B \r\n" );
document.write( "we can build C from A and B\r\n" );
document.write( "We can build D from C\r\n" );
document.write( "We can build E from A and B\r\n" );
document.write( "We can build F from D and E\r\n" );
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document.write( "So we start with this:\r\n" );
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document.write( "        p  q | (p^q) | ~(p^q) | (pvq) | ~(p^q) v (pvq) |\r\n" );
document.write( "case 1  T  T |       |        |       |                |\r\n" );
document.write( "case 2  T  F |       |        |       |                |\r\n" );
document.write( "case 3  F  T |       |        |       |                |\r\n" );
document.write( "case 4  F  F |       |        |       |                |\r\n" );
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document.write( "We fill the (p^q) column using the rule for \"and\".\r\n" );
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document.write( "Case 1 has a T under p and a T under q.\r\n" );
document.write( "That's both T's, so the (p^q) column gets a T, for case 1.\r\n" );
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document.write( "Case 2 has a T under p and a F under q,\r\n" );
document.write( "That's not both T's, so the (p^q) column gets a F for case 2.\r\n" );
document.write( "\"And\" needs 2 T's to be true.\r\n" );
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document.write( "Case 3 has a F under p and a T under q,\r\n" );
document.write( "That's not both T's, so the (p^q) column gets a F for case 3.\r\n" );
document.write( "\"And\" needs 2 T's to be true.\r\n" );
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document.write( "Case 4 has a F under p and a F under q,\r\n" );
document.write( "That's certainly not both T's, so the (p^q) column gets a F for case 4.\r\n" );
document.write( "\"And\" needs 2 T's to be true.\r\n" );
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document.write( "So the cases under (p^q) go \"TFFF\"\r\n" );
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document.write( "        p  q | (p^q) | ~(p^q) | (pvq) | ~(p^q) v (pvq) |\r\n" );
document.write( "case 1  T  T |   T   |        |       |                |\r\n" );
document.write( "case 2  T  F |   F   |        |       |                |\r\n" );
document.write( "case 3  F  T |   F   |        |       |                |\r\n" );
document.write( "case 4  F  F |   F   |        |       |                |  \r\n" );
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document.write( "Now to fill in the next column, we notice it has \"~\" or \"not\" before\r\n" );
document.write( "(p^q), so we put the exact opposite of what is in the (p^q) column.  The\r\n" );
document.write( "(p^q) column has \"TFFF\", so the ~(p^q) column has \"FTTT\":\r\n" );
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document.write( "        p  q | (p^q) | ~(p^q) | (pvq) | ~(p^q) v (pvq) |\r\n" );
document.write( "case 1  T  T |   T   |    F   |       |                |\r\n" );
document.write( "case 2  T  F |   F   |    T   |       |                |\r\n" );
document.write( "case 3  F  T |   F   |    T   |       |                |\r\n" );
document.write( "case 4  F  F |   F   |    T   |       |                |  \r\n" );
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document.write( "Next we fill in the (pvq) column.\r\n" );
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document.write( "Case 1 has a T under p and a T under q.\r\n" );
document.write( "All \"v\" needs is at least 1 T on either or both sides of it. p has a T\r\n" );
document.write( "and Q has a T, so that's at least one T on at least one side of \"v\", so\r\n" );
document.write( "case 1 gets a T.\r\n" );
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document.write( "Case 2 has a T under p and an F under q.\r\n" );
document.write( "All \"v\" needs is at least 1 T on either or both sides of it. p has a T\r\n" );
document.write( "and Q has a F, so that's at least one T on at least one side of \"v\", so\r\n" );
document.write( "case 2 gets a T.\r\n" );
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document.write( "Case 3 has a F under p and a T under q.\r\n" );
document.write( "All \"v\" needs is at least 1 T on either or both sides of it. p has a F\r\n" );
document.write( "and Q has a T, so that's at least one T on at least one side of \"v\", so\r\n" );
document.write( "case 3 gets a T.\r\n" );
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document.write( "Case 4 has a F under p and a F under q.\r\n" );
document.write( "But \"v\" needs at least 1 T on either or both sides of it to be true. It does\r\n" );
document.write( "NOT have T on either side of it in case 4, so case 4 gets an F.\r\n" );
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document.write( "So the (pvq) column goes TTTF\r\n" );
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document.write( "        p  q | (p^q) | ~(p^q) | (pvq) | ~(p^q) v (pvq) |\r\n" );
document.write( "case 1  T  T |   T   |    F   |   T   |                |\r\n" );
document.write( "case 2  T  F |   F   |    T   |   T   |                |\r\n" );
document.write( "case 3  F  T |   F   |    T   |   T   |                |\r\n" );
document.write( "case 4  F  F |   F   |    T   |   F   |                |  \r\n" );
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document.write( "Now we just have one more column to fill in.  It has \"v\" or \"OR\" between\r\n" );
document.write( "what's in the previous two columns, ~(p^q), (pvq).  All we need in order to\r\n" );
document.write( "put a true in that column is for just at least ONE of the preceding two\r\n" );
document.write( "columns to have a T in it. \r\n" );
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document.write( "Case 1 has a F under ~(p^q) and a T under (pvq),\r\n" );
document.write( "That's at least one T, so the last column gets a T, for case 2.\r\n" );
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document.write( "Case 2 has a T under ~(p^q) and a T under (pvq),\r\n" );
document.write( "That's at least one T, so the last column gets a T for case 2.\r\n" );
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document.write( "Case 3 has a T under ~(p^q) and a T under (pvq),\r\n" );
document.write( "That's at least one T, so the last column gets a T for case 3.\r\n" );
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document.write( "Case 4 has a T under ~(p^q) and an F under (pvq),\r\n" );
document.write( "That's at least one T, so the last column gets a T for case 4. \r\n" );
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document.write( "So the final column gets filled in TTTT.\r\n" );
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document.write( "That means the statement is ALWAYS true in EVERY case.  Since that happened\r\n" );
document.write( "we say ~(p^q) v (pvq) is ALWAYS true regardless of whether p or q is true or\r\n" );
document.write( "false.  That's called a \"tautology\".\r\n" );
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document.write( "By the way, the fancy word for \"and\" (\"^\") is \"conjunction.\r\n" );
document.write( "The fancy word for \"or\" (\"v\") is \"disjunction\".\r\n" );
document.write( "The fancy word for \"not\" (\"~\") is \"negation\".\r\n" );
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document.write( "Edwin</PRE> <BR>\n" );
document.write( "   </SPAN>\n" );
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