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I'm learning about sets and I've come across this question.

{x} ∈ {{x}}

I have to answer whether this statement is true or not. I put down true but I'm not 100% sure. If anyone can clear up what the two sets of brackets mean and if I was correct that would be great. Thanks

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  • $\begingroup$ True. {{$x$}} = {$x$} . The extra pair of brackets makes no sense in this case $\endgroup$
    – slhulk
    Nov 22, 2015 at 19:00
  • $\begingroup$ I believe it means that the element x belongs to the set of elements x. {x} being the element x and {{x}} being the set of elements x. $\endgroup$
    – Fede Poncio
    Nov 22, 2015 at 19:01
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    $\begingroup$ @slhulk you're mistaken: $\{\{x\}\} \ne \{x\}$ $\endgroup$
    – luka5z
    Nov 22, 2015 at 19:04
  • $\begingroup$ You should also ask... is $x\in\{\{x\}\}$? $\endgroup$
    – JP McCarthy
    Nov 22, 2015 at 19:08
  • $\begingroup$ x∈{{x}} isn't that also true? $\endgroup$
    – SearchingFor
    Nov 22, 2015 at 19:17

3 Answers 3

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See it this way: $x$ is an object. Then $\{x\}$ is a set consisting of this single object $x$. Such a set is called a singleton. Finally $X:=\bigl\{\{x\}\bigr\}$ is a set of sets, containing the single set $\{x\}$. As a set $X$ is a singleton as well. The question now is whether this $X$ contains the set $\{x\}$ as an element; and this is obviously true.

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For any expression $A$, "$\{A\}$" means "the set whose only element is $A$". In particular, when $A=\{x\}$, $\{\{x\}\}$ means the set whose only element is $\{x\}$. That is, the double brackets have no special meaning; they are just two pairs of single brackets which happen to be nested.

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The curly brackets $\{\}$denotes a set. So you have two sets, one $\{x\}$ and another $\{\{x\}\}$, lets denote the first set as $A=\{x\}$ , then the second set is $\{A\}$ and your statement is whether $A\in\{A\}$ or not, which is of course true.

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  • $\begingroup$ Thanks Michael, this made it a lot clearer. $\endgroup$
    – SearchingFor
    Nov 22, 2015 at 19:05

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