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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?

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    $\begingroup$ 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. $\endgroup$ – Regret Jan 17 '15 at 13:12
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    $\begingroup$ "Subset of" is not the same as "member of." $\endgroup$ – Thomas Andrews Jan 17 '15 at 13:16
  • $\begingroup$ Why isn't $\{\{a\}\}=\{a\}$? $\endgroup$ – Thomas Andrews Jan 17 '15 at 13:17
  • $\begingroup$ @ThomasAndrews are they the same? $\endgroup$ – shinzou Jan 17 '15 at 13:18
  • $\begingroup$ No, @kuhaku: $\{a\} \in \{\{a\}\}$, but $\{a\}\neq \{\{a\}\}$. $\endgroup$ – Namaste Jan 17 '15 at 13:20
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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:

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

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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.

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