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  1. Suppose $S$ is a subset of cardinality $c$. Given two elements $x,y \in S$, prove that there exist two disjoint subsets $S_1$ and $S_2$ of $S$ each of cardinality $c$ such that $x \in S_1, y \in S_2$.

  2. For two sets $S$ and $T$, prove that $|S| ≤ |T|$ implies $|\mathcal{P(S)}|≤|\mathcal{P(T)}|$.

  3. Let $\mathcal{P_0}(S)$ denote the collection of all countable subsets of $S$. Given that $|S| = |T| = c$, show that $|\mathcal{P_0}(S)| = |\mathcal{P_0}(T)|$.

Can someone please prove this for me? I'm really having a tough time with cardinality and don't know where to begin with. I'm assuming everyone is familiar with the $\mathcal{P}$ notation for the power set.

HKT
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    *powerset (or power set) not "power of a set" – parsiad Apr 07 '17 at 02:27
  • @parsiad Thanks. I've edited that. As you can see, I could do with your help. – HKT Apr 07 '17 at 02:28
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    Is $c$ supposed to be the continuum? The statement (1) is certainly not true for finite sets. It should also read "distinct" elements. – parsiad Apr 07 '17 at 02:28
  • @parsiad Yes, exactly. Are you sure? This is given as a challenge question and is most likely true. – HKT Apr 07 '17 at 02:28
  • That's not how Math.SE should work. What did you try? In 1, what does it mean that $\lvert S\rvert=\lvert \mathbb R\rvert$? How can you use that? In 2, what did you try? Unpack the definitions of what you're given and what you're trying to show. At what point do you get stuck? – martin.koeberl Apr 07 '17 at 02:35
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    *ALL* of theses three, unrelated questions, have been asked before. You're expected to ask one question per post. You're expected to search before you ask. – Asaf Karagila Apr 07 '17 at 04:22

1 Answers1

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Hint: If $S$ has cardinality of the continuum, there exists a bijection from $S$ to $\mathbb{R}$, call it $f$. Let $x,y$ be distinct elements of $S$. Pick $\delta>0$ such that $f(y)\notin B$ where $B\equiv(f(x)-\delta,f(x)+\delta)$. What can we say about $f^{-1}(B)$ and $f^{-1}(\mathbb{R}\setminus B)$?

Hint: If $S$ and $T$ are sets with $|S|\leq|T|$, we can find an injective function from $S$ to $T$, call it $f$. For an arbitrary subset $A$ of $S$, define $f(A)\equiv\{f(a)\colon a\in A\}$. What can we say about the mapping $A\mapsto f(A)$?

parsiad
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  • When people ask several different questions on the same post, they invariably cheat (knowingly or unknowingly) the daily question limit. Instead of posting an answer, you should suggest they post these as three different questions (which will make the duplicate closures make more sense). You could also vote to close as too broad. – Asaf Karagila Apr 07 '17 at 04:37
  • There was an additional question asked even after I posted my answer, it seems. – parsiad Apr 07 '17 at 04:54