The image shows two questions related to set theory. Let's analyze each one:

Question 1: $\emptyset \in \{\emptyset, \{\emptyset\}\}$

This asks whether the empty set ($\emptyset$) is an element of the set $\{\emptyset, \{\emptyset\}\}$.

• The set $\{\emptyset, \{\emptyset\}\}$ contains two elements: one is $\emptyset$ (the empty set itself), and the other is $\{\emptyset\}$ (a set containing the empty set).
• The symbol $\in$ checks for membership.
• Since $\emptyset$ is listed as an element of this set, the statement is True.

Question 2: $\emptyset \subseteq \{\emptyset, \{\emptyset\}\}$

This asks whether the empty set ($\emptyset$) is a subset of the set $\{\emptyset, \{\emptyset\}\}$.

• A set $A$ is a subset of a set $B$ if every element of $A$ is also an element of $B$.
• The empty set is a subset of every set, including $\{\emptyset, \{\emptyset\}\}$, because the empty set has no elements, so this condition is vacuously true.
• Therefore, this statement is True.

Let me know if you need any further explanation or details!

Here are 5 related questions to deepen understanding:

1. What is the difference between $\in$ and $\subseteq$ in set theory?
2. Can $\{\{\emptyset\}\} \subseteq \{\emptyset, \{\emptyset\}\}$ be true? Why or why not?
3. Is $\{\emptyset, \{\emptyset\}\} \subseteq \{\emptyset\}$ true or false?
4. How would the result change if the question was $\{\emptyset\} \in \{\emptyset, \{\emptyset\}\}$?
5. Can a set be both an element and a subset of another set?

Tip: In set theory, always pay attention to whether the question asks for membership ($\in$) or subset ($\subseteq$) relationships, as they test different concepts!