Let $E$ be a countably infinite set and consider the sets $F(E)$:=the set of all finite subsets of $E$ , and $I(E)$:=the set of all infinite subsets of $E$ , then what would be the nature of $F(E)$ and $I(E)$ (countable or uncountable ) ?, that is 'what is the set of all finite subsets of a countably infinite set , is it countable or uncountable?' and similarly 'what is the set of all infinite subsets of a countably infinite set , is it countable or uncountable?' . We notice that as $E$ is countable and infinite , $P(E)$ the power set of $E$ is uncountable and as $F(E) ∪ I(E)=P(E) $ , so at least one of $F(E)$ and $I(E)$ must be uncountable.
$ADDENDUM$:- What is the nature of all uncountable subsets of an uncountable set , is it countable or uncountable ?
Best Answer
HINT: Think of $E$ as being the natural number $\mathbb{N}$. Then there is a map from the power set of $\mathbb{N}$ to the open interval $(0,1)$ expressed in binary that takes a subset and maps it to a string $.a_1a_2a_3\ldots$ where the $n$th digit, $a_n$, is a $1$ if $n$ is in the subset and $0$ otherwise. If the subset is finite, then $.a_1a_2a_3\ldots$ will represent a rational number.