November 15th, 2011, 07:30 PM  #1 
Newbie Joined: Jun 2011 Posts: 21 Thanks: 0  Prove or Disprove functions
Let f:A>B , g:B>C , and are functions with gof:A>C (g composite f). Prove or disprove a.)if gof is 11 then f is 11 b) gof is onto , g is onto c.) if gof is 11 then g is 11 d.) If gof is onto, f is onto I started with a.) and i drew pictures to show myself its true, it looks like a transitive relationship, but how would write a mathematical proof that these are true? Please help. thank you 
November 17th, 2011, 08:57 AM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Prove or Disprove functions
a) "f is 1 to 1" means "if f(x)= f(y) then x= y". Since you want to prove that f is 1 to 1, start with f(x)= f(y)= z. Then g(f(x))= g(f(y))= g(z) so, since gf is 1 to 1, x= y. b) "g is onto" means that for any y in C there exist x in B such that g(x)= y. To prove that g is onto, let y be in C. Since gf is onto, there exist z in A such that gf(z)= y. Then x= f(z) is in B such that g(x)= y. 
November 22nd, 2011, 05:19 AM  #3 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Prove or Disprove functions
For (c) and (d), let A= {a, b,}, B= {p, q, r}, C= {x, y} and then f(a)= p, f(b)= q, g(p}= x, g(q)= g(r)= y. 

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