Let A, B, and C be sets, and let f:AarrowB and g:BarrowC be funtions. Prove or dis prove each of the following

Question

If the composite function g o f: AarrowC is a surjection, then the function f:AarrowB is a surjection.

If the composite function g o f:AarrowC is a surjection, then the function g:BarrowC is a surjection.

Answer 1

Let A = B = all non negative integers (it includes 0, although this does not matter).

Let C be the set {0,1}

f(a) = a - 10*INT(a/10)
in words: f(a) is the remainder when a s divided by 10.
if a is written in base 10, then f(a) is the lst digit.

example a = 734, then f(a) = 4

g(b) = b - 2*INT(b/2)
in words: f(b) is the remainder when b is divided by 2.
if b is even, g(b)=0; if b is odd, g(b)=1

Both functions are well-defined.

f:A-->B is not surjective,
yet g o f:A--->C is surjective.

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To prove the second one, I'd try to prove that g is not surjective even though g o f is surjective. This will lead to a contradiction.