Why does the curly bracket do not equal to the double curly brackets?

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$\{a\} \neq \{\{a\}\}$

$\{a\}$ is the set whose only element is the a (and no others). $\{\{a\}\}$ is the set whose only element is the set $\{a\}$.

Does this mean the 'element a' is not equal to 'set $\{a\}$'?

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3 Answers

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Even though people sometimes get sloppy about it, $a$ and $\{a\}$ are not the same object. $a$ is the only element of the set $\{a\}$.

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They are not equal.

Intuitively, $\{a\}$ means a set which contains an element $a$; while $\{\{a\}\}$ means a set that contains a set $\{a\}$ as its element.

From ZFC axiom: Every non-empty set $x$ contains a member $y$ such that $x$ and $y$ are disjoint sets.

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In general: $$\{x\}=\{y\}\iff x=y$$

Then we can conclude that also:$$\{x\}\neq\{y\}\iff x\neq y$$

Applying that in your case we find that the statement $\{a\}\neq\{\{a\}\}$ is the same statement as $a\neq\{a\}$.

Sidenote:

If also the axiom of regularity is accepted then this statement is true for every $a$.

This because on base of that axiom it can be proved that $a\notin a$ is true for every $a$ while $a=\{a\}$ implies that $a\in a$.

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