document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=192700>Question 192700</A>: <I><SPAN STYLE=\"word-break: break-all;\">  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #144637 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=144637\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "Let p, q, and r be the following statements: <BR>\n" );
document.write( "p: It is raining.<BR>\n" );
document.write( "q: The clouds are dark.<BR>\n" );
document.write( "r:  The temperature is dropping\r<BR>\n" );
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document.write( "A quick note: \r<BR>\n" );
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document.write( "the symbol \"~\" means \"not\". Ex: ~p: NOT p\r<BR>\n" );
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document.write( "the symbol \"^\" means \"and\". Ex: p ^ q: p AND q\r<BR>\n" );
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document.write( "the symbol \"v\" means \"or\". Ex: p v q: p OR q\r<BR>\n" );
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document.write( "the symbol \"-->\" means \"implies\" or stands for \"If..then\". Ex: p --> q: p IMPLIES q or IF p then Q\r<BR>\n" );
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document.write( "So this means that...\r<BR>\n" );
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document.write( "(a) p ^ r \r<BR>\n" );
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document.write( "Simply translates to: It is raining AND the temperature is dropping.\r<BR>\n" );
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document.write( "-------\r<BR>\n" );
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document.write( "(b) ~p ^ (q V r ) \r<BR>\n" );
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document.write( "Since \"v\" means \"or\", this means that q v r means \"the clouds are dark OR the temperature is dropping\". \r<BR>\n" );
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document.write( "Since \"^\" means \"and\" we get the final translation:\r<BR>\n" );
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document.write( "It is NOT raining AND either the clouds are dark OR the temperature is dropping.\r<BR>\n" );
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document.write( "(c) q --> r\r<BR>\n" );
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document.write( "Looking above, we see that \"-->\" means \"If..Then\". So in this case, this means that \"If q, then r\"\r<BR>\n" );
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document.write( "So the given statement translates to: IF the clouds are dark, THEN the temperature is dropping\r<BR>\n" );
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document.write( "(d) ( ~ r ^ ~ q) --> ~ p\r<BR>\n" );
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document.write( "First, negate r to get ~r. This translates to \"temperature is NOT dropping\"\r<BR>\n" );
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document.write( "Now, negate q to get ~q. This translates to \"the clouds are NOT dark\"\r<BR>\n" );
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document.write( "Negate p to get ~p. This translates to \"it is NOT raining\"\r<BR>\n" );
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document.write( "Combine ~r and ~q with a \"^\" (which means \"and\") to get: \"the temperature is NOT dropping AND the clouds are NOT dark\"\r<BR>\n" );
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document.write( "Now connect the last statement with ~p with a \"-->\" (which means \"If..then\") to get the final translation:\r<BR>\n" );
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document.write( "IF temperature is NOT dropping AND the clouds are NOT dark, THEN it is NOT raining\r<BR>\n" );
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document.write( "Note: all capitalized words are for emphasis (which are not normally capitalized)\r<BR>\n" );
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document.write( "============================================================\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "p: A dog is friendly.<BR>\n" );
document.write( "q: A dog has a long tail.<BR>\n" );
document.write( "r: A dog licks faces.\r<BR>\n" );
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document.write( "a) If a dog has a long tail, then it is not friendly.\r<BR>\n" );
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document.write( "First, we see that p: A dog is friendly\r<BR>\n" );
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document.write( "To get the statement \"the dog is NOT friendly\", simply negate \"p\" to get ~p\r<BR>\n" );
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document.write( "Now to translate a statement in the \"if..then\" form, simply use the \"-->\" to get: \r<BR>\n" );
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document.write( "Translation: q --> ~ p\r<BR>\n" );
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document.write( "b) If a dog is not friendly, then it does not lick faces.\r<BR>\n" );
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document.write( "Looking above, we see that \r<BR>\n" );
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document.write( "p: A dog is friendly.<BR>\n" );
document.write( "r: A dog licks faces.\r<BR>\n" );
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document.write( "So this makes\r<BR>\n" );
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document.write( "~p: A dog is NOT friendly<BR>\n" );
document.write( "~r: A dog does NOT lick faces\r<BR>\n" );
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document.write( "Since the statement is in the form \"If...then...\" this means that we'll use the \"-->\" again to get:\r<BR>\n" );
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document.write( "Translation: ~p --> ~ r\r<BR>\n" );
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document.write( "c) If a dog has a long tail, then either the dog is friendly or the dog licks faces\r<BR>\n" );
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document.write( "From the statements,\r<BR>\n" );
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document.write( "p: A dog is friendly.<BR>\n" );
document.write( "q: A dog has a long tail.<BR>\n" );
document.write( "r: A dog licks faces.\r<BR>\n" );
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document.write( "we can see that \"the dog is friendly or the dog licks faces\" translates to \r<BR>\n" );
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document.write( "p v r (where \"v\" means \"or\")\r<BR>\n" );
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document.write( "So \"If a dog has a long tail, then either the dog is friendly or the dog licks faces\" will utilize both the \"v\" and the \"-->\" operators to get:\r<BR>\n" );
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document.write( "Translation: q --> (p v r)\r<BR>\n" );
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document.write( "d) If a dog licks faces, then the dog is friendly and the dog has a long tail.\r<BR>\n" );
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document.write( "The consequent: \"the dog is friendly and the dog has a long tail\" translates to \r<BR>\n" );
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document.write( "p ^ q (where \"^\" means \"and\")\r<BR>\n" );
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document.write( "Because the statement is a conditional, this means that we'll use the operator \"-->\" to get:<BR>\n" );
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document.write( "Translation: r --> (p ^ q)\r<BR>\n" );
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document.write( "================================================================================\r<BR>\n" );
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document.write( "# 3 \r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "First, let's draw out the table for P ^ Q:\r<BR>\n" );
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document.write( "<tr><TD >P</TD><TD >Q</TD><TD >P ^ Q</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD></tr>\n" );
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document.write( "Take note how the last column P ^ Q has the first row of T and the rest of the rows are F. If we negate the third column (ie switch EVERY element from T to F and vice versa), we'll get:\r<BR>\n" );
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document.write( "<TABLE  border=\"1\">\n" );
document.write( "<tr><TD >P</TD><TD >Q</TD><TD >~(P ^ Q)</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD></tr>\n" );
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document.write( "which means that the column heading for the third column is: ~(p ^ q)\r<BR>\n" );
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document.write( "Note: you can use DeMorgan's Law to simplify ~(p ^ q) to get ~p v ~q, but this is optional.\r<BR>\n" );
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document.write( "b) \r<BR>\n" );
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document.write( "The truth table for P --> Q is :\r<BR>\n" );
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document.write( "<tr><TD >P</TD><TD >Q</TD><TD >P --> Q</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD></tr>\n" );
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document.write( "Now negate the third column to get:\r<BR>\n" );
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document.write( "<tr><TD >P</TD><TD >Q</TD><TD >~(P --> Q)</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD></tr>\n" );
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document.write( "So the compound statement for the third column is: ~(p --> q)\r<BR>\n" );
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document.write( "note: these type of exercises require you to be a bit creative or get lucky. After a while, you'll start to see the patterns. If you are willing to go the hard way, then try every possible truth table combination you can think of (ie try p v q, p ^ q, p --> q, ~(p v q), ~(p ^ q), ~(p --> q), etc...) \r<BR>\n" );
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document.write( "================================================================================\r<BR>\n" );
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document.write( "# 4\r<BR>\n" );
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document.write( "As a reminder, conditional statements are of the form \"If P, then Q\" where P is the antecendent and Q is the consequent.\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "So looking at the conditional \"If I go to work, then I will get paid\", we get\r<BR>\n" );
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document.write( "Antecedent: I go to work<BR>\n" );
document.write( "Consequent: I will get paid\r<BR>\n" );
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document.write( "The converse of \"If P, then Q\" is \"If Q, then P\". So the converse of \"If I go to work, then I will get paid.\" is: \r<BR>\n" );
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document.write( "Converse: \"If I get paid, then I will go to work\"\r<BR>\n" );
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document.write( "The inverse of \"If P, then Q\" is \"If NOT P, then NOT Q\". So the inverse of \"If I go to work, then I will get paid.\" is:\r<BR>\n" );
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document.write( "Inverse: \"If I do NOT go to work, then I will NOT get paid\"\r<BR>\n" );
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document.write( "Finally, the contrapositive of \"If P, then Q\" is \"If NOT Q, then NOT P\". \r<BR>\n" );
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document.write( "So the contrapositive of \"If I go to work, then I will get paid.\" is:\r<BR>\n" );
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document.write( "Contrapositive: \"If I do NOT get paid, then I will NOT go to work.\"\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Moving onto the conditional \"Your yard will not look good if you don’t mow the grass\", we can rearrange it to read \"IF you don't mow the grass, THEN your yard will not look good\". Take note it is now in the form \"If P then Q\" \r<BR>\n" );
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document.write( "So we can see that:\r<BR>\n" );
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document.write( "Antecedent: You don’t mow the grass<BR>\n" );
document.write( "Consequent: Your yard will not look good\r<BR>\n" );
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document.write( "The converse of \"If P, then Q\" is \"If Q, then P\". So the converse of \"If you don't mow the grass, then your yard will not look good.\" is: \r<BR>\n" );
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document.write( "Converse: \"If your yard will not look good, then you don't mow the grass\"\r<BR>\n" );
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document.write( "The inverse of \"If P, then Q\" is \"If NOT P, then NOT Q\". So the inverse of \"If you don't mow the grass, then your yard will not look good.\" is:\r<BR>\n" );
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document.write( "Inverse: \"If you mow the grass, then your yard looks good\"\r<BR>\n" );
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document.write( "Finally, the contrapositive of \"If P, then Q\" is \"If NOT Q, then NOT P\". \r<BR>\n" );
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document.write( "So the contrapositive of \"If you don't mow the grass, then your yard will not look good.\" is:\r<BR>\n" );
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document.write( "Contrapositive: \"If your yard looks good, then you will mow the grass.\"\r<BR>\n" );
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document.write( "# 5\r<BR>\n" );
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document.write( "One of DeMorgan's Laws states that\r<BR>\n" );
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document.write( "~(A v B) = ~A ^ ~B\r<BR>\n" );
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document.write( "where A and B are statements. In other words, if I negate something like: \"I work OR I play\" I'll get \"I do NOT work AND I do NOT play\" (ie I do neither). So a general rule with DeMorgan's Laws is to negate EVERYTHING (including the operators)\r<BR>\n" );
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document.write( "So in this case, ~(p v ~q) = ~p ^ ~~q  (the right reads: NOT p AND NOT NOT q)\r<BR>\n" );
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document.write( "Since NOT NOT cancels out (think of it as a double negative), this means that we can rewrite ~~q as just q\r<BR>\n" );
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document.write( "So \r<BR>\n" );
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document.write( "~(p v ~q) = ~p ^ ~~q = ~p ^ q\r<BR>\n" );
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document.write( "or simply\r<BR>\n" );
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document.write( "~(p v ~q) = ~p ^ q\r<BR>\n" );
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document.write( "# 6\r<BR>\n" );
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document.write( "Let \r<BR>\n" );
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document.write( "p = it is raining<BR>\n" );
document.write( "q = we will close the window\r<BR>\n" );
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document.write( "So the statement \"If it is raining, then we will close the window\", translates to \r<BR>\n" );
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document.write( "p --> q\r<BR>\n" );
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document.write( "So the argument...\r<BR>\n" );
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document.write( "If it is raining, then we will close the window. <BR>\n" );
document.write( "We closed the window.<BR>\n" );
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document.write( ":. It is raining\r<BR>\n" );
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document.write( "translates to:\r<BR>\n" );
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document.write( "p --> q<BR>\n" );
document.write( "q<BR>\n" );
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document.write( ":. p\r<BR>\n" );
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document.write( "Now the question is: is this argument valid?\r<BR>\n" );
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document.write( "To find out, let's make a truth table. Make the columns \"p\" and \"q\" followed by a column for \"p --> q\" and then columns for \"q\" and \"p\"\r<BR>\n" );
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document.write( "<tr><TD >p</TD><TD >q</TD><TD >p --> q</TD><TD >q</TD><TD >p</TD></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr>\n" );
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document.write( "Note: the entries in the column \"p --> q\" are all true except when you have p is true AND q is false (ie \"p --> q\" is false when p is true AND q is false).\r<BR>\n" );
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document.write( "Now the 3rd and 4th columns are the premises and the last column is the conclusion. If we find a row with all true premises but with a false conclusion, then the argument is invalid. Take a look at the 3rd row: since the premises for \"p --> q\" and \"q\" are true but the conclusion \"p\" is false, this leads to an invalid argument.\r<BR>\n" );
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document.write( "================================================================================\r<BR>\n" );
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document.write( "# 7\r<BR>\n" );
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document.write( "First, let's draw a circle denoting the set of all helpful people. Since \"All doctors are helpful\", this means that the set of doctors is a subset of the set of helpful people. So visually, the set \"doctors\" is a smaller circle completely contained in the set \"helpful people\".\r<BR>\n" );
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document.write( "Now because \"Some doctors are female\", this means that the circle to represent the set of females will overlap the set of doctors like so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/eulerdiagram1.png\">\r<BR>\n" );
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document.write( "Now because EVERY doctor is a helpful person and some doctors are females, this means that the set \"females\" and \"helpful people\" MUST overlap in some way. This overlap is in green:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/eulerdiagram2.png\">\r<BR>\n" );
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document.write( "So in this region, there are helpful people who are female. So this supports the conclusion that \"Some helpful people are female\" (given the previous premises).\r<BR>\n" );
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document.write( "So this shows that the given argument is valid\r<BR>\n" );
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document.write( "================================================================================\r<BR>\n" );
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document.write( "# 8\r<BR>\n" );
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document.write( "To construct the truth table for ~q  --> (~q ^ p), let's first break it up:\r<BR>\n" );
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document.write( "First, let's make a truth table for ~q:\r<BR>\n" );
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document.write( "<TABLE  border=\"1\">\n" );
document.write( "<tr><TD >q</TD><TD >~q</TD></tr><tr><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD></tr>\n" );
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document.write( "Note: just write the opposite of q for every entry\r<BR>\n" );
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document.write( "Now, let's make a table for ~q ^ p (simply use the \"and\" operator for columns ~q and p):\r<BR>\n" );
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document.write( "<TABLE  border=\"1\">\n" );
document.write( "<tr><TD >p</TD><TD >q</TD><TD >~q</TD><TD >~q ^ p</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "Finally, make a table for ~q  --> (~q ^ p)\r<BR>\n" );
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document.write( "<TABLE  border=\"1\">\n" );
document.write( "<tr><TD >p</TD><TD >q</TD><TD >~q</TD><TD >~q ^ p</TD><TD >~q  --> (~q ^ p)</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr>\n" );
document.write( "</TABLE> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );