Let f : A to B and g : B to C.

if g o f is surjective then g must be surjective

how do you prove this

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- Dec 12th 2009, 03:08 PMleinadwerdnaproof about composition
Let f : A to B and g : B to C.

if g o f is surjective then g must be surjective

how do you prove this - Dec 12th 2009, 03:18 PMtonio
- Dec 12th 2009, 04:33 PMleinadwerdna
- Dec 12th 2009, 04:54 PMJameson

Since g o f is surjective and is a transformation from A->C, any c in C must have at least one element a in A such that g o f(a) = c. It could have more, but it has at least one.

Now let b be in B and be the result of f(a), thus f(a)=b. Since g o f(a) = c is surjective, g(b) is by substitution. So for every c in C, there exists b in B such that g(b)=c, thus g is a surjection.

That was just writing tonio's post into words. Hopefully it has no errors, but the idea is there.