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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 ?

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  • $\begingroup$ Isn't this a duplicate of math.stackexchange.com/questions/361320/…? $\endgroup$ – Gerry Myerson Sep 14 '13 at 12:49
  • $\begingroup$ @GerryMyerson: Not an exact duplicate since I asked the "set of all infinite subsets ..." counter part , but I've got my answer thanks for the reference . $\endgroup$ – Souvik Dey Sep 14 '13 at 12:53
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    $\begingroup$ For the addendum, let $X$ be an uncountable set. Then $X\setminus\{x\}$ is uncountable for each $x\in X$, so the bijection $x\mapsto X\setminus\{x\}$ immediately shows that $X$ has uncountably many uncountable sets. $\endgroup$ – Brian M. Scott Sep 14 '13 at 22:06
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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.

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