Does there exist an injection from $P(S)$ to $u(S)$

Question

Let $S$ be an uncountable set , let $u(S)$ denote the set of all uncountable subsets of $S$ and let $P(S)$ denote , as usual , the powerset i.e. the set of all subsets of $S$, then does there exist an injection $f:P(S)→u(S)$ ?

Answer 1

Yes. $|S\times\{0,1\}|=|S|$, so replace $u(S)$ by $u(S\times\{0,1\})$. If $A\subseteq S$ is countable, let $\varphi(A)=(S\times\{0\})\cup(A\times\{1\})$; otherwise let $\varphi(A)=A\times\{1\}$. (I am assuming the axiom of choice here.)