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I am bit confused with the concept of empty set here.

Since {} is a subset of every set, it is a subset of itself? and hence {} = {{}}?

Also, say A = {a}, but since {} is a subset of A, is it true that A = {a, {}}, if so, what is its cardinality?

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4 Answers

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There is a big difference between subsets and elements.

The empty set is a subset of every set, including itself. However, it is not an element of itself, which is what $\{\}=\{\{\}\}$ would mean.

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Being a subset of a set does not mean it is an element of a set.

The set $\{\{\}\}$ has an element, whereas $\{\}$ does not have any elements. Therefore the sets are distinct (recall that two sets are equal if and only if they have the same elements).

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As many other's have pointed out, the empty set is a subset of every set, it is not an element of every set.

$\{\} \subseteq A$ is always true for any set $A$.

$\{\} \in A$ is not always true.

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The set $\{\}$ has no elements. The set $\{\{\}\}$ has one element, the empty set. Thus $\{\} \neq \{\{\}\}$

also, $A=\{a\} \neq \{a,\{\}\}$ for the very same reasons. The right hand side set has two elements: $a$ and $\{\}$, while $A$ has only $a$ as am element.

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