1. **Problem Statement:** Given the universal set $\varepsilon = \{x : -20 \leq x \leq 20\}$ and sets $M = \{x : -20 < x < 15\}$, $N = \{x : -10 < x \leq 10\}$, $P = \{x : 9 \leq x < 18\}$, find: a. $M'$ (complement of $M$) b. $N \cap P$ (intersection of $N$ and $P$) c. $n(N \cup P)$ (number of elements in union of $N$ and $P$) d. $N \cap M'$ (intersection of $N$ and complement of $M$) --- 2. **Formulas and Rules:** - Complement $A' = \varepsilon \setminus A$ - Intersection $A \cap B = \{x : x \in A \text{ and } x \in B\}$ - Union $A \cup B = \{x : x \in A \text{ or } x \in B\}$ - $n(A)$ is the count of elements in set $A$ --- 3. **Step-by-step Solutions:** **a. Find $M'$:** - $M = \{x : -20 < x < 15\}$ means $x$ is strictly between -20 and 15. - $\varepsilon = \{x : -20 \leq x \leq 20\}$ includes -20 and 20. - So, $M' = \varepsilon \setminus M = \{x : x \leq -20 \text{ or } x \geq 15\}$. - Since $\varepsilon$ only includes $-20 \leq x \leq 20$, $M' = \{-20, 15, 16, \ldots, 20\}$. **b. Find $N \cap P$:** - $N = \{x : -10 < x \leq 10\}$ - $P = \{x : 9 \leq x < 18\}$ - Intersection is $x$ values common to both. - Overlap is $9 \leq x \leq 10$. - So, $N \cap P = \{x : 9 \leq x \leq 10\}$. **c. Find $n(N \cup P)$:** - $N \cup P$ includes all $x$ in $N$ or $P$. - $N = \{x : -10 < x \leq 10\}$ - $P = \{x : 9 \leq x < 18\}$ - Union covers $-10 < x < 18$. - Since $x$ are integers (implied by set notation), count integers from -9 to 17 inclusive. - Count = $17 - (-9) + 1 = 27$. **d. Find $N \cap M'$:** - From (a), $M' = \{-20, 15, 16, \ldots, 20\}$ - $N = \{x : -10 < x \leq 10\}$ - Intersection is elements in both. - Since $M'$ includes only $x \leq -20$ or $x \geq 15$, and $N$ is between -10 and 10, no overlap. - So, $N \cap M' = \emptyset$ (empty set). --- **Final answers:** a. $M' = \{-20, 15, 16, 17, 18, 19, 20\}$ b. $N \cap P = \{9, 10\}$ c. $n(N \cup P) = 27$ d. $N \cap M' = \emptyset$