Suppose is a commutative diagram of $\displaystyle R$-modules and $\displaystyle R$-module homomorphisms.

(a) Suppose that $\displaystyle \phi$ is injective. Prove that the map $\displaystyle \phi_0 : \text{Ker } f \rightarrow \text{Ker } g$ is injective.

(b) Suppose that $\displaystyle \psi$ is surjective. Prove that the map $\displaystyle \bar{\psi} : \text{Coker } f \rightarrow \text{Coker } g$ is surjective.