The exercise in the image asks to represent the sets $X = \{1\}, Y = \{2, 3\}, Z = \{2, 5, 6\}$ with a Venn diagram and then solve the following problems:

1. $X \times Y$
2. $Y \times Z$
3. $X \times (Y \cap Z)$
4. $(X \times Y) \cap (X \times Z)$
5. $(Z - Y) \times (X \cup Y)$

Let's go step by step:

1. $X \times Y$

This is the Cartesian product of sets $X$ and $Y$, which means all possible ordered pairs where the first element is from $X$ and the second element is from $Y$.

Since $X = \{1\}$ and $Y = \{2, 3\}$, we get: $X \times Y = \{ (1, 2), (1, 3) \}$

2. $Y \times Z$

Similarly, the Cartesian product $Y \times Z$ gives all ordered pairs with the first element from $Y$ and the second from $Z$.

With $Y = \{2, 3\}$ and $Z = \{2, 5, 6\}$, we get: $Y \times Z = \{ (2, 2), (2, 5), (2, 6), (3, 2), (3, 5), (3, 6) \}$

3. $X \times (Y \cap Z)$

Now, we need to find the intersection $Y \cap Z$, which is the set of elements common to both $Y$ and $Z$.

Since $Y = \{2, 3\}$ and $Z = \{2, 5, 6\}$, we have: $Y \cap Z = \{2\}$ Thus, $X \times (Y \cap Z) = X \times \{2\} = \{ (1, 2) \}$.

4. $(X \times Y) \cap (X \times Z)$

First, we calculate $X \times Z$: $X \times Z = \{ (1, 2), (1, 5), (1, 6) \}$ Now, we take the intersection of $X \times Y$ and $X \times Z$. These two sets are: $X \times Y = \{ (1, 2), (1, 3) \}$ $X \times Z = \{ (1, 2), (1, 5), (1, 6) \}$ The only common element is $(1, 2)$, so: $(X \times Y) \cap (X \times Z) = \{ (1, 2) \}$

5. $(Z - Y) \times (X \cup Y)$

First, we calculate $Z - Y$, which is the set difference (elements in $Z$ that are not in $Y$).

Since $Z = \{2, 5, 6\}$ and $Y = \{2, 3\}$, we have: $Z - Y = \{5, 6\}$

Next, we compute $X \cup Y$, the union of $X$ and $Y$, which includes all elements in either set: $X \cup Y = \{1, 2, 3\}$

Finally, we find the Cartesian product: $(Z - Y) \times (X \cup Y) = \{ (5, 1), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3) \}$

Would you like more details on any of these steps? Here are five questions that expand on this problem:

1. What are the key differences between union, intersection, and Cartesian product of sets?
2. How would the solution change if one of the sets was empty?
3. Can the Cartesian product be applied to more than two sets simultaneously?
4. How would we visualize these set relationships using a Venn diagram?
5. How does set difference differ from symmetric difference?

Tip: When working with sets, it's helpful to first compute any intersections or differences before applying operations like Cartesian products to simplify the problem.