# Is it true that $\emptyset\notin\{\emptyset,\{\emptyset\}\}$

I've been having a hard time understanding why $\emptyset\notin\{\emptyset,\{\emptyset\}\}$. Why isn't it an element in that set? Also why is $\{\emptyset\}\in\{\emptyset\}$ true? I know that $\{0\}\notin\{0\}$ so why is that one true and this false?

Determine whether these statements are true or false.

1. $\emptyset\in\{\emptyset\}$
2. $\emptyset\in\{\emptyset,\{\emptyset\}\}$
3. $\{\emptyset\}\in\{\emptyset\}$
4. $\{\emptyset\}\in\{\{\emptyset\}\}$
5. $\{\emptyset\}\subset\{\emptyset,\{\emptyset\}\}$
6. $\{\{\emptyset\}\}\subset\{\emptyset,\{\emptyset\}\}$
7. $\{\{\emptyset\}\}\subset\{\{\emptyset\},\{\emptyset\}\}$
• Can you provide more context? To me it seems the other way around, i.e. $\emptyset \in \{\emptyset, \{\emptyset\}\}$ but $\{\emptyset\} \not\in \{\emptyset\}$. – el_tenedor Apr 9 '17 at 9:07
• I believe so too, the answers given were wrong. Thanks! – Tala Salim Apr 9 '17 at 9:14

Let's move through each statement one by one.

• $\emptyset\in\{\emptyset\}$

This must be true, as the elements in $\{\emptyset\}$ are everything in between in braces, namely $\emptyset$.

• $\emptyset\in\{\emptyset,\{\emptyset\}\}$

This is also true, for the same reason. The elements of $\{\emptyset,\{\emptyset\}\}$ are $\emptyset$ and $\{\emptyset\}$.

• $\{\emptyset\}\in\{\emptyset\}$

This is false. The element in $\{\emptyset\}$ is just $\emptyset$, and $\emptyset\ne\{\emptyset\}$.

• $\{\emptyset\}\in\{\{\emptyset\}\}$

This is true. The element in $\{\{\emptyset\}\}$ is $\{\emptyset\}$, so $\{\emptyset\}$ is in $\{\{\emptyset\}\}$.

• $\{\emptyset\}\subset\{\emptyset,\{\emptyset\}\}$

This is true. The statement is asking whether or not all of the element in the set on the left are also all elements of the set on the right. Since the element in $\{\emptyset\}$ is $\emptyset$, and $\emptyset\in\{\emptyset,\{\emptyset\}\}$, it's true!

• $\{\{\emptyset\}\}\subset\{\emptyset,\{\emptyset\}\}$

This is true. Again, we want to know if $\{\emptyset\}\in\{\emptyset,\{\emptyset\}\}$, since $\{\emptyset\}$ is the only element in the set $\{\{\emptyset\}\}$. Because $\{\emptyset\}$ is in $\{\emptyset,\{\emptyset\}\}$, this implies that $\{\{\emptyset\}\}\subset\{\emptyset,\{\emptyset\}\}$.

• $\{\{\emptyset\}\}\subset\{\{\emptyset\},\{\emptyset\}\}$

This one is a little tricky. We should know that sets cannot contain duplicates, so $\{1,1,2,2,3\}=\{1,2,3\}$. In this example, that means that the question is actually asking if $\{\{\emptyset\}\}\subset\{\{\emptyset\}\}$. Since $\{\{\emptyset\}\}=\{\{\emptyset\}\}$, this is false, since $\subset$ implies proper containment, that is, that the two sets are not equal.

For these questions, it can be helpful to rewrite the questions with different elements. For example, let $\emptyset = 0$ and $\{\emptyset\}=1$. Then $\{\emptyset\}\subset\{\emptyset,\{\emptyset\}\}$ becomes $\{0\}\subset\{0,1\}$.

• This was extremely helpful, thank you so much! – Tala Salim Apr 10 '17 at 19:02
• Glad I could help! – Santana Afton Apr 10 '17 at 20:58

It is true that $\emptyset \in \{ \emptyset, \{\emptyset\}\}$. It is false that $\{ \emptyset \} \in \{ \emptyset \}$. You're right to have been suspicious of both those results.

• Do you mind checking if the answers to the picture attached is a)T b)T c)F d)T e)T f)F g)F I'm not sure about the last two but I know I can't be trusting the answers given – Tala Salim Apr 9 '17 at 9:14