Let's analyze each of the statements to determine if they are true or false and provide explanations where necessary.

1. $x \in \{x, y, z\}$

• Interpretation: This statement asserts that $x$ is an element of the set $\{x, y, z\}$.
• True or False: True.
• Explanation: Since the set $\{x, y, z\}$ contains the element $x$, the statement $x \in \{x, y, z\}$ is true.

2. $x \subseteq \{x, y, z\}$

• Interpretation: This statement asserts that $x$ is a subset of the set $\{x, y, z\}$.
• True or False: False.
• Explanation: For $x$ to be a subset of $\{x, y, z\}$, $x$ itself must be a set where every element of $x$ is also in $\{x, y, z\}$. However, $x$ is not a set; it is an element. Thus, this statement is false.

3. $\{x\} \in \{x, y, z\}$

• Interpretation: This statement asserts that the set $\{x\}$ (a set containing only $x$) is an element of the set $\{x, y, z\}$.
• True or False: False.
• Explanation: The set $\{x\}$ is not one of the elements in the set $\{x, y, z\}$, which contains only the elements $x$, $y$, and $z$. The set $\{x\}$ is different from the element $x$.

4. $x \subseteq \{\{x\}, \{y\}, \{z\}\}$

• Interpretation: This statement asserts that $x$ is a subset of the set $\{\{x\}, \{y\}, \{z\}\}$.
• True or False: False.
• Explanation: For $x$ to be a subset of $\{\{x\}, \{y\}, \{z\}\}$, $x$ would need to be a set where each of its elements is one of $\{x\}, \{y\}, \{z\}$. But $x$ is not a set, so it cannot be a subset of another set. Hence, this statement is false.

5. $x \in \{\{x\}, \{y\}, \{z\}\}$

• Interpretation: This statement asserts that $x$ is an element of the set $\{\{x\}, \{y\}, \{z\}\}$.
• True or False: False.
• Explanation: The set $\{\{x\}, \{y\}, \{z\}\}$ contains three elements: $\{x\}$, $\{y\}$, and $\{z\}$. It does not contain $x$ directly as an element, only as a part of these sets. Therefore, the statement is false.

Summary

1. True
2. False
3. False
4. False
5. False

Would you like more details on any of these explanations, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. What is the difference between an element and a subset in set theory?
2. Can a set be an element of another set? Provide an example.
3. How would you determine if one set is a proper subset of another?
4. What does it mean for an element to belong to a set?
5. How does the concept of "power sets" relate to these kinds of problems?

Tip: Always remember that in set theory, elements and subsets are distinct concepts. An element is a member of a set, while a subset is a set that contains some or all members of another set.