1. **State the problem:** We have two sets of homework problems chosen by students: - $X$: problems 1 to 7 (application problems) - $Y$: problems 8 to 10 (challenge problems) We want to determine which statements correctly describe the compound events $X \cap Y$ (intersection) and $X \cup Y$ (union). 2. **Recall definitions:** - The intersection $X \cap Y$ is the set of outcomes common to both $X$ and $Y$. - The union $X \cup Y$ is the set of outcomes in either $X$ or $Y$ or both. 3. **Analyze the sets:** - $X = \{1,2,3,4,5,6,7\}$ - $Y = \{8,9,10\}$ 4. **Find $X \cap Y$:** Since $X$ and $Y$ have no common elements (numbers 1–7 vs. 8–10), $$X \cap Y = \emptyset$$ 5. **Find $X \cup Y$:** $$X \cup Y = \{1,2,3,4,5,6,7,8,9,10\}$$ 6. **Evaluate each statement:** - A: "Choosing number 5 is part of $X \cap Y$." Number 5 is in $X$ but not in $Y$, so not in the intersection. **False**. - B: "Choosing number 7 is part of $X \cup Y$." Number 7 is in $X$, so in the union. **True**. - C: "Choosing number 9 is part of $X \cap Y$." Number 9 is in $Y$ but not in $X$, so not in the intersection. **False**. - D: "The total number of outcomes in $X \cup Y$ is 0." The union has 10 elements, so **False**. - E: "The total number of outcomes in $X \cap Y$ is 0." The intersection is empty, so size is 0. **True**. **Final correct answers:** B and E.