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

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?

## marked as duplicate by Andrés E. Caicedo, Yiyuan Lee, Shuchang, user85798, Thomas AndrewsMar 29 '14 at 4:20

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.

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).

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.

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.