Determine whether a given function is a pseudorandom generator/function I am trying to solve the following three tasks (for exam practice, not as a homework):


Define $ : \{0,1\}^* \rightarrow \{0,1\}^*$ by $G(x_1,...,x_n) = _1 \oplus _2,_1,⋅⋅⋅,x_n.$ Prove that this $G$ is not a pseudorandom generator.
Let $G$ be a pseudorandom generator, and define $G'(x_1, ..., x_n) = G(x_1, ..., x_n)|(x_1 \vee x_2)$. Is $G'$ a pseudorandom generator?
Define $ : \{0,1\}^* × \{0,1\}^* \rightarrow \{0,1\}^*$ as follows: $_{_1,...,_} (_1,..., _) = \bigoplus_i _ _$, where $_, _ \in \{0,1\}^*$ (Note that, different from the usual convention, $F$ takes an n-bit key and an n-bit input, but has only a single-bit output). Prove that this  is not a pseudorandom function.


My guesses are:


Not a pseudorandom generator since a distinguisher can always distinguish G(s) from a truly random string because the first bit of G(s) is always equal to the XOR of the second and third bit.
Not a pseudorandom generator because a distinguisher could simply check whether the last bit of the string equals $x_1 \vee x_2$ and could thus distinguish G(s) from a truly random string.
I cannot determine how to solve this.


Are my guesses correct? Can someone provide an idea for 3?
I have fully solved the questions now.


Not a pseudorandom generator since the first bit of $G(s)$ is always equal to the XOR of the second and third bit, i.e. a distinguisher can easily tell $G(s)$ apart from a truly random string $r$.
Not a pseudorandom generator. We can for example construct a distinguisher $D$ that, on input of a string $w$, outputs 1 if and only if the final bit is 0. If $w$ is uniformly distributed then the final bit is 0 with probability $\frac{1}{2}$ but if $w = G(s)$ for a uniformly distributed seed s  the final bit will be 0 with probability $\frac{1}{4}$.
Not a pseudorandom function. A distinguisher D could tell $F_k$ apart from a truly random function $f$ in the following way: Given access to an oracle $W$, D queries $W(0...0)$. If $W = F_k$ then the result will always be 0, but if $W$ is a random function then it should be 0 only with probability $\frac{1}{2}$.