1. **Problem statement:** Given the universal set $U$ as the months of the year, and sets $J$, $Y$, $V$, and $R$ defined by certain letter conditions, find: - $n(J \cup Y)$ - $n(J \cap V)$ - $n(R')$ 2. **Step 1: Define the sets explicitly.** - $U = \{\text{January, February, March, April, May, June, July, August, September, October, November, December}\}$ - $J = \{x \in U \mid x \text{ begins with } J\} = \{\text{January, June, July}\}$ - $Y = \{x \in U \mid x \text{ ends with } Y\} = \{\text{January, February, May, July}\}$ - $V = \{x \in U \mid x \text{ begins with a vowel}\} = \{\text{January, April, August, October}\}$ - $R = \{x \in U \mid x \text{ ends with } R\} = \{\text{October, December}\}$ 3. **Step 2: Find $n(J \cup Y)$.** - $J = \{\text{January, June, July}\}$ - $Y = \{\text{January, February, May, July}\}$ - The union $J \cup Y$ includes all elements in either $J$ or $Y$: $$J \cup Y = \{\text{January, June, July, February, May}\}$$ - Count the elements: $$n(J \cup Y) = 5$$ 4. **Step 3: Find $n(J \cap V)$.** - $J = \{\text{January, June, July}\}$ - $V = \{\text{January, April, August, October}\}$ - The intersection $J \cap V$ includes elements in both $J$ and $V$: $$J \cap V = \{\text{January}\}$$ - Count the elements: $$n(J \cap V) = 1$$ 5. **Step 4: Find $n(R')$.** - $R = \{\text{October, December}\}$ - The complement $R'$ is all elements in $U$ not in $R$: $$R' = U \setminus R = \{\text{January, February, March, April, May, June, July, August, September, November}\}$$ - Count the elements: $$n(R') = 10$$ **Final answers:** - $n(J \cup Y) = 5$ - $n(J \cap V) = 1$ - $n(R') = 10$