If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

Question

If the empty set is a subset of every set, why it isn't written with the elements of a set? like so $\{1,2,3,\emptyset\}$

Or why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements and the other has 1 but since the empty set is a subset of both, then why it isn't being mentioned explicitly in the definition of the set?

The empty set is a subset of every set. Not an element of every set. Recall the definition of subset and you will see why this is so. — Regret, Jan 17 '15 at 13:12
${{a}}$ is the set with one element: ${a}$, an element which happens also to be a set. Be careful though: Note that $a\notin {{a}}$. What we do have is that $a\in {a},$ but $a \notin {{a}}$. — amWhy, Jan 17 '15 at 13:22
No, I was arguing by analogy. ${\emptyset}$ is not equal to $\emptyset$ in the same way that ${{a}}$ is not equal to ${a}$. @kuhaku — Thomas Andrews, Jan 17 '15 at 13:46

score 10 · Accepted Answer · answered Jan 17 '15 at 13:15

The empty set is a subset of every set, but it is not an element of every set.

In your examples, $$\varnothing \in \{\varnothing, \{a\}\}\text{ but } \varnothing \notin \{\{a\}\},$$

but it is a subset of both sets:

$$\varnothing \subset \{\varnothing, \{a\}\}, \text{ and } \varnothing \subset \{\{a\}\}.$$

score 3 · Answer 2 · answered Jan 17 '15 at 13:18

What happens if $\varnothing$ is in fact an element of some set? For example, $\{\varnothing\}$?

How can you tell the difference between $\varnothing$ and $\{\varnothing\}$, if you add the empty set as an element to every set?

Membership and inclusion are two different relations, and should be treated differently.

If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

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