Question

Let S, T, U be three non-void sets and f : S $\to$ T,g : T $\to$ U is surjective. Then

• A. g and f are both surjective
• B. g is surjective, f may not be so
• C. f is surjective, g may not be so
• D. f and g both may not be surjective

Answer 1

Answer: (B)

g is surjective, f may not be so

Answer 2

g o f : S $\to$ U is onto Let z be an arbitrary element of U $\because$ g o f : S $\to$ U onto there exists x $\in$ S g o f(x) = z $\Rightarrow$ g(f(x)) = z;g(y) = z , where y = f(x) $\in$ T for all z $\in$ U, there exists y = f(x) $\in$ T such that g(y) = z g : T $\to$ U onto.