identify if the functions are pseudorandom generator? Given a pseudorandom generator (PRG) with $G(x) = y_{1} || y_{2}$, where $y_{1}$ and $y_{2}$ are n-bit strings. $G'(x) := G(y_{1}) || G(y_{2})$. Is $G'(.)$ still a PRG?
I am guessing it's a No because $G'(x) := G(y_{1}) || G(y_{2})$ always outputs the concatenation of $G(x)$ with previously generated outputs as inputs. But I am not sure what's the formal way of explaining this.

In addition. $G''(x) = G(y_{1}) ||G(y_{1})\oplus G(y_{2})|| G(y_{2})$, is $G''(x)$ still a PRG? I would say No again in this case because the pattern is again the concatenation of $G'(x)$ with the $\oplus $ in the middle?

I think I might be entirely wrong on this. A pointer or two would be appreciated.
$G'$ is a PRG. Proof sketch: by assumption, $G(x)$ for a uniform $x$ is indistinguishable from a uniform string. This means that $y_1$ and $y_2$ (the halves of $G(x)$) are also indistinguishable from uniform strings, and finally that so are $G(y_1)$ and $G(y_2)$.