document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103170>Question 1103170</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f : A ——> B and g : B ——> C , two bijective functions. Prove that g•f is bijective </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717956 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717956\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let a, b be elements of A such that \r<BR>\n" );
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document.write( "g(f(a)) = g(f(b)).\r<BR>\n" );
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document.write( "g 1-to-1 ==> f(a) = f(b).\r<BR>\n" );
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document.write( "f 1-to-1 ==> a = b.\r<BR>\n" );
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document.write( "==> g(f(x)) is 1-to-1 also.-------------(1)\r<BR>\n" );
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document.write( "Now let z be an element of C.\r<BR>\n" );
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document.write( "g  DISABLED_event_onto= => there is y an element of B such that g(y) = z.\r<BR>\n" );
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document.write( "f  DISABLED_event_onto= => there is x an element of A such that f(x) = y. \r<BR>\n" );
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document.write( "===> g(f(x)) = z.\r<BR>\n" );
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document.write( "===> g o f is also onto.--------------(2)\r<BR>\n" );
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document.write( "Therefore g o f is also bijective, from (1) and (2). </SPAN>\n" );
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