Cardinality of Sets and Infinite Sets The following are homework questions I would like assistance on. I will do what I can to work on these problems; any feedback is helpful.
In the following problems, S is an infinite set (we do not know if it is denumerable or uncountable).
Question 1:
Let $k$ be in P.
Define $G_k(S)$ = {$A$: $A$ is an element of $\mathcal{P}(S))$, |$A$|=$k$ }
Show that |$G_k(S)$| = |$S$|.
My intuition tells me that I will need to use Cantor's Theorem and the Schroeder-Bernstein Theorem but I am having difficulty beginning the proof.
Question 2:
Let $C$ be a denumerable collection of sets and for every $T$ in C, T is equipotent to S.
Show that $|\bigcup C| = |S|$
Question 3:
Let $F(S)$ $=$ {$A$: $A$ is an element of $\mathcal{P}(S)$, $S$ \ $A$ is finite}
Show that |$F(S)$| = |$S$|

The first question is simply false without assuming the axiom of choice, so you can use that. You could also use the fact that if $S$ is infinite then $S\times S$ is equipotent with $S$. Therefore $S^k$ is equipotent with $S$. Note that in particular cases (e.g. $\mathbb N$ or $\mathbb R$) this is true even if the axiom of choice is not assumed. However for the general case this requires the axiom of choice to hold.
Now find an injection from $S$ into $G_k(S)$; and find a surjection from $S^k$ onto $G_k(S)$, from the above we know that we can now find a surjection from $S$ onto $G_k(S)$. Again, using the axiom of choice we can therefore find an injective function from $G_k(S)$ into $S$, and use Cantor-Bernstein.
For the second question you need the fact, $|S\times\mathbb N|=|S|$. Then write $C=\{T_n\mid n\in\mathbb N\}$, and use the fact that every $T_n$ has a bijection with $S$. Again, this requires the axiom of choice.
Lastly, the third question is equivalent to showing that $Fin(S)=\{A\subseteq S\mid |A|<\infty\}$ is equipotent with $S$; and this follows from the previous question. You could write $Fin(S)$ as the countable union of $G_k(S)$.
