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

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**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**.

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**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**.

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

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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?

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

∅ ∈ {∅, {∅}} and ∅ ⊆ {∅, {∅}}

Explore how the empty set ∅ relates to the set {∅, {∅}} through element membership and subset relations. Learn about key set theory concepts, including the definition of subsets and membership in undergraduate-level mathematics.

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