Let's analyze the problem in the image:

You are given two sets:

• $X = \{ 1, 2, 3, 5 \}$
• $Y = \{ 3, 5, 6 \}$

The two tasks are:

1. Find $(Y \cap X) \times Y$ (the Cartesian product of the intersection of $Y$ and $X$ with $Y$).
2. Find $n(Y^2)$ (the number of elements in $Y^2$).

1. $(Y \cap X) \times Y$

First, calculate the intersection $Y \cap X$:

• $Y \cap X = \{ 3, 5 \}$ (the elements common to both sets).

Now, calculate the Cartesian product $(Y \cap X) \times Y$, which means forming all ordered pairs $(a, b)$, where $a \in Y \cap X$ and $b \in Y$:

• $(Y \cap X) = \{ 3, 5 \}$
• $Y = \{ 3, 5, 6 \}$

The Cartesian product is: $\{ (3,3), (3,5), (3,6), (5,3), (5,5), (5,6) \}$

2. $n(Y^2)$

$Y^2$ refers to the Cartesian product of $Y \times Y$, meaning all ordered pairs $(a, b)$, where both $a$ and $b$ are elements of $Y$. The size $n(Y^2)$ is the number of elements in the Cartesian product $Y \times Y$.

The Cartesian product $Y \times Y$ will be: $Y \times Y = \{ (3,3), (3,5), (3,6), (5,3), (5,5), (5,6), (6,3), (6,5), (6,6) \}$

So, $n(Y^2) = 9$, since there are 9 ordered pairs.

Summary:

1. $(Y \cap X) \times Y = \{ (3,3), (3,5), (3,6), (5,3), (5,5), (5,6) \}$
2. $n(Y^2) = 9$

Would you like more details or further explanations on any part of the solution?

Here are 5 related questions you might find useful:

1. What are the properties of the Cartesian product of two sets?
2. How does the intersection of two sets behave in set theory?
3. Can you generalize the formula for the size of a Cartesian product for larger sets?
4. What is the difference between set union and set intersection?
5. How do you calculate the power set of a given set?

Tip: When performing a Cartesian product, always ensure that each element from the first set is paired with every element of the second set.