1. **Problem statement:** We are given sets of months and need to find the cardinalities of certain unions, intersections, and complements. 2. **Recall the universal set and subsets:** - $U = \{\text{January, February, March, April, May, June, July, August, September, October, November, December}\}$ (12 months) - $J = \{x \in U \mid x \text{ begins with } J\} = \{\text{January, June, July}\}$ - $Y = \{x \in U \mid x \text{ ends with } Y\} = \{\text{January, February, May, July}\}$ - $V = \{x \in U \mid x \text{ begins with a vowel}\} = \{\text{January, April, August, October}\}$ - $R = \{x \in U \mid x \text{ ends with } R\} = \{\text{October, December}\}$ 3. **Find $n(J \cup Y)$:** - $J \cup Y$ includes all months that start with J or end with Y. - $J = \{\text{January, June, July}\}$ - $Y = \{\text{January, February, May, July}\}$ - Union: $J \cup Y = \{\text{January, June, July, February, May}\}$ - Count: $n(J \cup Y) = 5$ 4. **Find $n(J \cap V)$:** - Intersection means months that start with J and also start with a vowel. - $J = \{\text{January, June, July}\}$ - $V = \{\text{January, April, August, October}\}$ - Intersection: $J \cap V = \{\text{January}\}$ - Count: $n(J \cap V) = 1$ 5. **Find $n(R')$:** - $R' = U \setminus R$ means months not ending with R. - $R = \{\text{October, December}\}$ - So $R' = \{\text{January, February, March, April, May, June, July, August, September, November}\}$ - Count: $n(R') = 10$ **Final answers:** - $n(J \cup Y) = 5$ - $n(J \cap V) = 1$ - $n(R') = 10$