Which of the following are true? Give a proof for true statements and a countere

Question

Which of the following are true? Give a proof for true statements and a counterexample for false ones: The compostion of two surjective functions is surjective.

Explanation / Answer

true Suppose that f : A -->-> B and g : B -->-> C are onto. Then we want to show that g o f : A --> C is onto. So, take any c E C. Since g is onto, there is an element b E B so that g(b) = c. Since f is onto, there is an element a E A. so that f(a) = b. But then (g o f)(a) = g(f(a)) = g(b) = c Since we stated with an arbitrary element c E C and we found an element a E A which maps onto it, we have shown that g o f is surjective.

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