The question in the image involves set theory and algebraic notation. Let's break it down step by step:

Given Sets:

• $X = \{3, 4\}$
• $Y = \{4, 5\}$
• $Z = \{5, 6\}$

The tasks are:

1. $X \times (Y \cap Z)$
2. $Y^2$
3. $n(X^2)$

Solution:

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

• Intersection of Y and Z: $Y \cap Z = \{4, 5\} \cap \{5, 6\} = \{5\}$
• Cartesian product: $X \times \{5\} = \{(3, 5), (4, 5)\}$

2. $Y^2$

• This notation $Y^2$ seems unclear, but in set theory, it could mean the Cartesian product $Y \times Y$. If this is the case: $Y \times Y = \{(4, 4), (4, 5), (5, 4), (5, 5)\}$ So, $Y^2 = Y \times Y$.

3. $n(X^2)$

• $X^2$ probably refers to $X \times X$, the Cartesian product of $X$ with itself: $X \times X = \{(3, 3), (3, 4), (4, 3), (4, 4)\}$
• Number of elements in $X^2$: $n(X^2) = 4$

Would you like further details on any part of this solution?

Here are 5 related questions to expand your understanding:

1. What is the union of sets $X$, $Y$, and $Z$?
2. How does the Cartesian product differ from the union of sets?
3. Can you compute $(Y \cup Z) \times X$?
4. What is the power set of $X$?
5. How would the intersection of $X$ and $Z$ be computed?

Tip: Cartesian products produce pairs of elements from two sets, making the result much larger than simple set operations like union or intersection.