From the image, we are solving problems related to set operations. Here's a breakdown of the given information and the required operations:

Given Sets:

• $U = \{1, 2, 3, 4, 5\}$ (Universal set)
• $X = \{1, 2, 4\}$
• $Y = \{1, 3, 4\}$

Operations to Solve:

1. $X \cap Y$ (Intersection)
2. $X \cup Y$ (Union)
3. $\overline{Y}$ (Complement of $Y$ in $U$)
4. $X \cap \overline{Y}$ (Intersection of $X$ and $\overline{Y}$)

Solutions:

1. $X \cap Y$: The intersection of $X$ and $Y$ is the set of elements common to both: $X \cap Y = \{1, 2, 4\} \cap \{1, 3, 4\} = \{1, 4\}$ Correct answer: $\{1, 4\}$.

2. $X \cup Y$: The union of $X$ and $Y$ is the set of all elements in either $X$ or $Y$: $X \cup Y = \{1, 2, 4\} \cup \{1, 3, 4\} = \{1, 2, 3, 4\}$ Correct answer: $\{1, 2, 3, 4\}$.

3. $\overline{Y}$: The complement of $Y$ in $U$ is the set of elements in $U$ but not in $Y$: $\overline{Y} = U \setminus Y = \{1, 2, 3, 4, 5\} \setminus \{1, 3, 4\} = \{2, 5\}$ Correct answer: $\{2, 5\}$.

4. $X \cap \overline{Y}$: First, we use $\overline{Y} = \{2, 5\}$. Then, find the intersection of $X$ and $\overline{Y}$: $X \cap \overline{Y} = \{1, 2, 4\} \cap \{2, 5\} = \{2\}$ Correct answer: $\{2\}$.

Let me know if you want further clarifications or details! 😊

Here are 5 related questions to deepen understanding:

1. What is the difference between a union and an intersection in set theory?
2. How is the complement of a set determined relative to the universal set?
3. If $X = \{a, b, c\}$ and $Y = \{b, c, d\}$, what is $X \cup Y$?
4. Can the intersection of two sets be an empty set? Provide an example.
5. Why is it important to specify the universal set when finding complements?

Tip:

Always list the universal set explicitly in set theory problems to avoid ambiguity when computing complements.