document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=521599>Question 521599</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that:<BR>\n" );
document.write( "(a) P ) Q is not equivalent to Q ) P<BR>\n" );
document.write( "(b) P ) Q is not equivalent to :P ) :Q<BR>\n" );
document.write( "(c) P , Q , R is not equivalent to (P , Q) ^ (Q , R)<BR>\n" );
document.write( " Give shorter propositions which are equivalent to:<BR>\n" );
document.write( "(a) T , T<BR>\n" );
document.write( "(b) T , F<BR>\n" );
document.write( "(c) P , T.<BR>\n" );
document.write( "(d) P , F.\r<BR>\n" );
document.write( "\n" );
document.write( "In a mathematics book we read<BR>\n" );
document.write( "y = x2 + 2x + 2 ) y = (x + 1)2 + 1 ) y >_ 1: <BR>\n" );
document.write( "that ist >_ littl equal<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 #346546 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=346546\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ x^2\ +\ 2x\ +\ 2\ \Rightarrow\ y\ =\ (x\ +\ 1)^2\ +\ 1\ \Rightarrow\ y\ \geq\ 1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ x^2\ +\ 2x\ +\ 2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "is a concave up parabola such that the vertex is a minimum point and where the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex is equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{-2}{2}\ =\ -1\"> and the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-1)^2\ +\ 2(-1)\ +\ 2\ =\ 1\">, the minimum value of the function is, indeed, 1 and the given proposition is true.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );