Functions

Let f:A\rightarrow B be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation \sim on the set A as
a_1\sim a_2 \text{ if } f(a_1)=f(a_2),
where a_1, a_2 \in A . Let \varepsilon =\{[x]:x \in A\} be the set of all the equivalence classes under \sim . Define a new mapping F: \varepsilon \rightarrow B as
F([x]) = f(x), for all the equivalence classes [x] in \varepsilon
Which of the following statements is/are TRUE?

Which one of the following is the closed form for the generating function of the sequence \{a_n \}_{n \geq 0} defined below?
a_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.

Consider the following sets, where n\geq 2:

S1: Set of all nxn matrices with entries from the set \{a, b, c\}
S2: Set of all functions from the set\{0,1,2,...,n^2-1\} to the set \{0, 1, 2 \}

Which of the following choice(s) is/are correct?
[MSQ]

If the ordinary generating function of a sequence \{a_{n}\}_{n=0}^{\infty} \; is \; \frac{1+z}{(1-z)^{3}} then a_{3}-a_{0} is equal to ______.

If g(x)=1-x and h(x)=\frac{x}{x-1}, then \frac{g(h(x))}{h(g(x))} is: