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The image presents a question asking whether certain set-theoretic statements are true or false, along with explanations for why. I will work through each one of the provided set-related problems.

The notation includes elements like `∈` (meaning "is an element of") and `⊂` (meaning "is a subset of"). Here’s the breakdown of each:

### 1. \( x \in \{x, y, z\} \)

- **True**: The element \( x \) is part of the set \( \{x, y, z\} \), as it is explicitly listed.

### 2. \( \{x\} \subseteq \{x, y, z\} \)

- **True**: The set \( \{x\} \) is a subset of \( \{x, y, z\} \), as all elements of \( \{x\} \) (in this case, just \( x \)) are contained within \( \{x, y, z\} \).

### 3. \( x \in \{\{x, y\}, z\} \)

- **False**: The element \( x \) is not part of the set \( \{\{x, y\}, z\} \). This set contains the set \( \{x, y\} \) and the element \( z \), but \( x \) itself is not an element of this set.

### 4. \( \{x, y\} \subseteq \{\{x\}, \{y\}, z\} \)

- **False**: The set \( \{x, y\} \) is not a subset of \( \{\{x\}, \{y\}, z\} \), because \( \{x, y\} \) contains elements \( x \) and \( y \), but the set on the right-hand side only contains \( \{x\} \), \( \{y\} \), and \( z \). The elements \( x \) and \( y \) are not directly present in the right-hand set.

### 5. \( x \in \{\{x\}, \{y\}, z\} \)

- **False**: The element \( x \) is not directly contained in the set \( \{\{x\}, \{y\}, z\} \). The set contains \( \{x\} \), which is a set that contains \( x \), but \( x \) itself is not an element of the outer set.

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Would you like any more details on these concepts?

Here are 5 related questions to expand on this:
1. What is the difference between an element of a set and a subset?
2. How do you determine whether a set is a subset of another?
3. Can a set be an element of another set? How does that differ from being a subset?
4. How does the power set relate to the subsets of a given set?
5. What are the different notations for set relations and their meanings?

**Tip:** When checking if a set is a subset, remember that all elements of the first set must be found in the second set exactly as they are.

A. Tell whether each of the following is true or false. If false, explain why: 1. x ∈ {x, y, z}, 2. {x} ⊆ {x, y, z}, 3. x ∈ {{x, y}, z}, 4. {x, y} ⊆ {{x}, {y}, z}, 5. x ∈ {{x}, {y}, z}

Analyze whether each set theory statement is true or false, focusing on set membership and subset relations. Understand the differences between elements and subsets, explained step-by-step.

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