In each part of this problem, give examples of sets A,B,C and functions f : A → B and g : B → C satisfying the indicated properties.
a) g is not injective but g ◦ f is injective.
b) f is not surjective but g ◦ f is surjective.
Suggestion: Work with sets having at most 3 elements.
For a.) Would F(1,2)=1 and G(1,2,3)=2,3 work?
For b.) Would F(1,2,3)=2,3 and G(1)=1 work?
Im looking for some feedback? And I cannot figure out the sets A,B,C?
Take $A= \big \{1,2 \big \}, B= \big \{1,2,3 \big \}, C= \big \{1,2 \big \}$. Define $f:A \rightarrow B, g:B \rightarrow C$ by \begin{align} f(1)=1, f(2)=2 \end{align} \begin{align} g(1)=1, g(2)=2, g(3)=3 \end{align} Then $g$ is not injective and $f$ is not surjective. And $g \circ f:A \rightarrow C$ is given by \begin{align} (g \circ f)(1) =g(f(1)) =g(1)=1 \\ (g \circ f)(2) =g(f(2)) =g(2)=2 \end{align} Note that $g \circ f$ is both injective and surjective.