document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=681470>Question 681470</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion. Show complete work and submit your solution to the Dropbox. \r<BR>\n" );
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document.write( "I) If the cat does not have claws, then the cat cannot scratch the furniture.<BR>\n" );
document.write( "II) If the cat can scratch the furniture, then the cat has claws.<BR>\n" );
document.write( "III) If the cat has claws, then the cat can scratch the furniture. <BR>\n" );
document.write( "a. I and II are equivalent <BR>\n" );
document.write( "b. I and III are equivalent<BR>\n" );
document.write( "c. II and III are equivalent <BR>\n" );
document.write( "d. I, II, and III are equivalent<BR>\n" );
document.write( "e. None are equivalent \r<BR>\n" );
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document.write( "(Points : 4) <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 #422672 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=422672\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the statements are:<BR>\n" );
document.write( "I) If the cat does not have claws, then the cat cannot scratch the furniture.<BR>\n" );
document.write( "II) If the cat can scratch the furniture, then the cat has claws.<BR>\n" );
document.write( "III) If the cat has claws, then the cat can scratch the furniture. <BR>\n" );
document.write( "you can solve this using truth tables.<BR>\n" );
document.write( "you would do so as follows:<BR>\n" );
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document.write( "p = cat has claws\r\n" );
document.write( "q = cat scratches furniture\r\n" );
document.write( "~p->~q = I\r\n" );
document.write( "q->p = II\r\n" );
document.write( "p->q = III\r\n" );
document.write( "truth table:\r\n" );
document.write( "             p   q   ~p   ~q   ~p->~q   q->p   p->q\r\n" );
document.write( "             T   T    F    F      T       T     T\r\n" );
document.write( "             T   F    F    T      T       T     F\r\n" );
document.write( "             F   T    T    F      F       F     T\r\n" );
document.write( "             F   F    T    T      T       T     T\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "from the truth tables, it appears that ~p->~q is equivalent to q->p because their truth tables are the same.<BR>\n" );
document.write( "that means that I is equivalent to II<BR>\n" );
document.write( "III is not equivalent to either one of them because its truth table is different.<BR>\n" );
document.write( "if we examine the statement, we will see why this is so.<BR>\n" );
document.write( "the statement are, once again:<BR>\n" );
document.write( "I) If the cat does not have claws, then the cat cannot scratch the furniture.<BR>\n" );
document.write( "II) If the cat can scratch the furniture, then the cat has claws.<BR>\n" );
document.write( "III) If the cat has claws, then the cat can scratch the furniture.<BR>\n" );
document.write( "statement II looks like it is the contrapositive of statement I.<BR>\n" );
document.write( "statement I says:<BR>\n" );
document.write( "if no claws, then no scratch furniture.<BR>\n" );
document.write( "statement II says:<BR>\n" );
document.write( "if scratch furniture, then claws.<BR>\n" );
document.write( "if the statement is:<BR>\n" );
document.write( "if no claws, then no scratch furniture<BR>\n" );
document.write( "then the contrapositive to that statement is:<BR>\n" );
document.write( "if scratch furniture, then claws.<BR>\n" );
document.write( "if the statement is:<BR>\n" );
document.write( "if scratch furniture, then claws<BR>\n" );
document.write( "then the contrapositive to that statement is:<BR>\n" );
document.write( "if no claws, then no scratch furniture.<BR>\n" );
document.write( "the statement and its contrapositive are equivalent which means if the statement is true then the contrapositive is also true and vice versa.<BR>\n" );
document.write( "this is the case between statements I and II.<BR>\n" );
document.write( "statement III says:<BR>\n" );
document.write( "if claws, then scratch furniture.<BR>\n" );
document.write( "the contrapositive to this would be.<BR>\n" );
document.write( "if no scratch furniture, then no claws.<BR>\n" );
document.write( "all 3 statements are:<BR>\n" );
document.write( "I = if no claws, then no scratch furniture<BR>\n" );
document.write( "II = if scratch furniture, then claws.<BR>\n" );
document.write( "III = if claws, then scratch furniture.<BR>\n" );
document.write( "the contrapositive to statement III is:<BR>\n" );
document.write( "if no scratch furniture, then no claws<BR>\n" );
document.write( "neither I or II are the contrapositive to statement III, therefore the only equivalent statement are I and II.<BR>\n" );
document.write( "It's a little convoluted, but that's the way it works out.<BR>\n" );
document.write( "Use of the truth tables can make sense of the insensible, assuming you know how to use them.<BR>\n" );
document.write( "p->q is only false if p is true and q is false.<BR>\n" );
document.write( "q->p is only false if q is true and p is false.<BR>\n" );
document.write( "~p->~q is only false if ~p is true and ~q is false.<BR>\n" );
document.write( "~q->~p is only false if ~q is true and ~p is false.<BR>\n" );
document.write( "that's the truth table for the implied statement.l\r<BR>\n" );
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document.write( "  </SPAN>\n" );
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