document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=25126>Question 25126</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f:X to Y and g:Y to W such that g.f is injective.<BR>\n" );
document.write( "a) Show f is injective.<BR>\n" );
document.write( "b) Give an example of functions f and g for which g.f is injective but g is not injective.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #13490 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=13490\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Let f:X to Y and g:Y to W such that g.f is injective.<BR>\n" );
document.write( "a) Show f is injective.<BR>\n" );
document.write( "b) Give an example of functions f and g for which g.f is injective <BR>\n" );
document.write( " but g is not injective.\r<BR>\n" );
document.write( "\n" );
document.write( " a) f(x) = f(y) --> g(f(x)) = g(f(y)) --> g.f(x) = g.f(y)<BR>\n" );
document.write( "   --> x = y (since g.f is injective)<BR>\n" );
document.write( " Hence, f is injecitve.\r<BR>\n" );
document.write( "\n" );
document.write( " b) Set f:{1} --> {1} to be f(x) = x (identity map) and<BR>\n" );
document.write( "    g:{1,2} --> {2} to be g(x) = 2 for x =1,2.<BR>\n" );
document.write( " We see that g.f:{1}--> {2 } is injective, but g is not.\r<BR>\n" );
document.write( "\n" );
document.write( " Kenny </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );