Exercise $2.$ Katz$-$Lindell Exercise $3\mathrm{.}6[40$ points$]$ Let $G$ be a pseudorandom

generator. For each of the following cases, determine whether $G^{\text{'}}$ is necessarily a

pseudorandom generator. If yes, give a proof; if not, provide a counterexample.

Define $G^{\text{'}}(s)=G(\frac{?}{\text{b}\text{a}\text{r}}(s)),$ where $\frac{?}{\text{b}\text{a}\text{r}}(s)$ is the complement of $s.$

Define $G^{\text{'}}(s)\frac{?}{b}=ar(G(s)).$

Define $G^{\text{'}}(s)=G(0^{|s|}||s).$

Define $G^{\text{'}}(s)=G(s)||G(s+1).$