1. **State the problem:** We have the universal set $U$ as the months of the year: \{January, February, March, April, May, June, July, August, September, October, November, December\}. Sets are defined as: - $J = \{x \in U \mid x \text{ begins with } J\}$ - $Y = \{x \in U \mid x \text{ ends with } Y\}$ - $V = \{x \in U \mid x \text{ begins with a vowel}\}$ - $R = \{x \in U \mid x \text{ ends with } R\}$ We need to find: 1. $n(V \cap R)$ 2. $n(R')$ 3. $n(J \cup Y)$ --- 2. **Identify the elements of each set:** - $U = \{$January, February, March, April, May, June, July, August, September, October, November, December$\}$ - $J = \{$January, June, July$\}$ (months starting with J) - $Y = \{$January, May, July$\}$ (months ending with Y) - $V = \{$January, April, August, October$\}$ (months starting with vowels A, E, I, O, U) - $R = \{$September, October, November, December$\}$ (months ending with R) --- 3. **Calculate $n(V \cap R)$:** Find months that start with a vowel and end with R. - $V = \{$January, April, August, October$\}$ - $R = \{$September, October, November, December$\}$ Intersection $V \cap R = \{$October$\}$ So, $n(V \cap R) = 1$ --- 4. **Calculate $n(R')$:** $R'$ is the complement of $R$ in $U$, i.e., months not ending with R. Since $n(U) = 12$ and $n(R) = 4$, $$n(R') = n(U) - n(R) = 12 - 4 = 8$$ --- 5. **Calculate $n(J \cup Y)$:** Union of $J$ and $Y$ includes all months that start with J or end with Y. - $J = \{$January, June, July$\}$ - $Y = \{$January, May, July$\}$ Union $J \cup Y = \{$January, June, July, May$\}$ So, $n(J \cup Y) = 4$ --- **Final answers:** $$n(V \cap R) = 1$$ $$n(R') = 8$$ $$n(J \cup Y) = 4$$