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.

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.