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

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
• "Subset of" is not the same as "member of." – Thomas Andrews Jan 17 '15 at 13:16
• Why isn't $\{\{a\}\}=\{a\}$? – Thomas Andrews Jan 17 '15 at 13:17
• @ThomasAndrews are they the same? – shinzou Jan 17 '15 at 13:18
• No, @kuhaku: $\{a\} \in \{\{a\}\}$, but $\{a\}\neq \{\{a\}\}$. – amWhy Jan 17 '15 at 13:20

## 2 Answers

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:

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

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.