composition of function

Question

Let f : A → B and g : B → C denote two functions. Consider the following two statements:
S₁ : If both f and g are injections then the composition function gof: A → C is an injection.
S₂ : If the function gof: A → C is surjection and g is an injection then the function f is a surjection.
S₃ : If h(a) = g(f(a)) and h(a) is onto then g must be onto, where ∀a, a ∈ A.
Which of the above statements are valid?

please give explanation

Comments

Detailed Video Solution, with PROOF: https://youtu.be/1qOTs03imLU

Answer 1

all are valid

Comments

3rd is not valid. while first two are valid

for the case of third:

A={a}

B={b ,c}

C={d,e}

f:(a,  b )

g:{(b,d),(b,e)}

gof is onto but f is not onto

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I think you have misinterpreted the question, he is asking whether g is a surjection or not.