1. **Rewrite the set** $A = \{x \mid x \in \mathbb{Z}, -5 < x < -2\}$ **in roster form.** The set $A$ contains all integers $x$ such that $x$ is greater than $-5$ and less than $-2$. Since $x$ must be an integer, the values are $-4$ and $-3$. So, $A = \{-4, -3\}$. 2. **Given sets:** $U = \{1,2,3,4,5,6,7,8,9,10\}$ $A = \{2,4,6,8,10\}$ $B = \{1,2,3,4,5\}$ $C = \{4,5,6,7\}$ **a. Calculate** $(A \cup B) - C$ - First, find $A \cup B$ (union of $A$ and $B$): $$A \cup B = \{1,2,3,4,5,6,8,10\}$$ - Then subtract $C$ (elements in $C$) from this union: $$ (A \cup B) - C = \{1,2,3,4,5,6,8,10\} - \{4,5,6,7\} = \{1,2,3,8,10\} $$ **b. Calculate** $(A \cap B) - C$ - Find $A \cap B$ (intersection of $A$ and $B$): $$A \cap B = \{2,4\}$$ - Subtract $C$: $$ (A \cap B) - C = \{2,4\} - \{4,5,6,7\} = \{2\} $$ **c. Calculate** $(A \cup B \cup C)'$ - First find $A \cup B \cup C$: $$A \cup B \cup C = \{1,2,3,4,5,6,7,8,10\}$$ - The complement is all elements in $U$ not in this union: $$ (A \cup B \cup C)' = U - (A \cup B \cup C) = \{9\} $$ **d. Calculate** $(A \cup B) \cap C$ - Recall $A \cup B = \{1,2,3,4,5,6,8,10\}$ - Intersection with $C$: $$ (A \cup B) \cap C = \{1,2,3,4,5,6,8,10\} \cap \{4,5,6,7\} = \{4,5,6\} $$ 3. **Pet store purchases problem:** - Number who purchased dog product: $83$ - Number who purchased cat product: $101$ - Number who purchased fish product: $22$ - Number who purchased both dog and cat products: $31$ This data can be used to analyze overlaps and total customers, but since no specific question is asked here, we stop at listing the data. **Final answers:** 1. $A = \{-4, -3\}$ 2a. $(A \cup B) - C = \{1,2,3,8,10\}$ 2b. $(A \cap B) - C = \{2\}$ 2c. $(A \cup B \cup C)' = \{9\}$ 2d. $(A \cup B) \cap C = \{4,5,6\}$