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

  • 1
    Shouldn't question 1 be to show $|G_k(S)|=|S|$?2012-11-07
  • 0
    Yes, you're right. Sorry about that!2012-11-07

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