$4\mathrm{.}1\mathrm{.}2.$ Problem. Let $S$ be any set.

$(1)$ Construct an infective map $$:Slongrightarrow$2^S.$

$(2)$ Prove that there is no bijection $$:Slongrightarrow$2^S$ from $S$ to $2^S.$

Remark $4\mathrm{.}1\mathrm{.}2.$ For the proofs,

You need to deal with the case $S=\frac{O}{?}$ separately.

Since there is an injection $$:Slongrightarrow$2^S$ but there is no bijection $$:Slongrightarrow$2^S,$ we see that #$S$#$2^S.$