1. **State the problem:** Given sets $U = \{1, 2, 3, 4, 5\}$, $X = \{2, 4\}$, and $Y = \{1, 3, 4\}$, find: - $X \cap Y$ - $X \cup Y$ - $\overline{X}$ (complement of $X$ relative to $U$) - $X \cap \overline{Y}$ 2. **Recall definitions:** - Intersection $A \cap B$ is the set of elements in both $A$ and $B$. - Union $A \cup B$ is the set of elements in $A$ or $B$ or both. - Complement $\overline{A}$ relative to $U$ is the set of elements in $U$ not in $A$. 3. **Calculate $X \cap Y$:** Elements common to $X$ and $Y$ are those in both sets. $X = \{2,4\}$ $Y = \{1,3,4\}$ Common element is $4$. So, $X \cap Y = \{4\}$. 4. **Calculate $X \cup Y$:** Union includes all elements from both sets without repetition. $X \cup Y = \{1, 2, 3, 4\}$. 5. **Calculate $\overline{X}$:** Complement of $X$ relative to $U$ is elements in $U$ not in $X$. $U = \{1, 2, 3, 4, 5\}$ $X = \{2, 4\}$ So, $\overline{X} = \{1, 3, 5\}$. 6. **Calculate $X \cap \overline{Y}$:** First find $\overline{Y}$: $Y = \{1, 3, 4\}$ $\overline{Y} = U \setminus Y = \{2, 5\}$ Now intersect with $X$: $X = \{2, 4\}$ $X \cap \overline{Y} = \{2\}$. **Final answers:** 1. $X \cap Y = \{4\}$ 2. $X \cup Y = \{1, 2, 3, 4\}$ 3. $\overline{X} = \{1, 3, 5\}$ 4. $X \cap \overline{Y} = \{2\}$