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 |$F(S)$| = |$S$|