4.I.1D

Let $A, B$ and $C$ be non-empty sets and let $f: A \rightarrow B$ and $g: B \rightarrow C$ be two functions. For each of the following statements, give either a brief justification or a counterexample.

(i) If $f$ is an injection and $g$ is a surjection, then $g \circ f$ is a surjection.

(ii) If $f$ is an injection and $g$ is an injection, then there exists a function $h: C \rightarrow A$ such that $h \circ g \circ f$ is equal to the identity function on $A$.

(iii) If $X$ and $Y$ are subsets of $A$ then $f(X \cap Y)=f(X) \cap f(Y)$.

(iv) If $Z$ and $W$ are subsets of $B$ then $f^{-1}(Z \cap W)=f^{-1}(Z) \cap f^{-1}(W)$.

*Typos? Please submit corrections to this page on GitHub.*