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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$\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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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