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November 15th, 2011, 07:30 PM   #1
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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 1-1 then f is 1-1

b) gof is onto , g is onto

c.) if gof is 1-1 then g is 1-1

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
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November 17th, 2011, 08:57 AM   #2
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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.
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November 22nd, 2011, 05:19 AM   #3
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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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