- 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 $x \in S_1, y \in S_2$.
For two sets $S$ and $T$, prove that $|S| ≤ |T|$ implies $|\mathcal{P(S)}|≤|\mathcal{P(T)}|$.
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.
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 $A\mapsto f(A)$?