1. **Problem statement:** Given sets \(\epsilon = \{x : -20 \leq x \leq 20\}\), \(M = \{x : -20 < x \leq 15\}\), \(N = \{x : -10 < x \leq 10\}\), and \(P = \{x : 9 \leq x < 18\}\), find: a. \(M'\) (complement of \(M\) in \(\epsilon\)) 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. **Formula and rules:** - The complement \(M'\) is all elements in \(\epsilon\) not in \(M\). - Intersection \(A \cap B\) is elements common to both sets. - Union \(A \cup B\) is all elements in either set. - \(n(S)\) is the number of elements in set \(S\). - Since sets are intervals of integers, count elements by counting integers in the interval. 3. **Step-by-step solutions:** a. Find \(M'\): - \(M = \{x : -20 < x \leq 15\}\) means all integers from \(-19\) to \(15\). - \(\epsilon = \{x : -20 \leq x \leq 20\}\) means integers from \(-20\) to \(20\). - So, \(M' = \epsilon \setminus M = \{-20\} \cup \{16,17,18,19,20\}\). b. Find \(N \cap P\): - \(N = \{x : -10 < x \leq 10\} = \{-9,-8,\ldots,10\}\). - \(P = \{x : 9 \leq x < 18\} = \{9,10,11,\ldots,17\}\). - Intersection is integers common to both: \(\{9,10\}\). c. Find \(n(N \cup P)\): - Union covers from \(-9\) to \(17\) because \(N\) goes up to 10 and \(P\) starts at 9. - So union is \(\{-9,-8,\ldots,17\}\). - Number of elements is \(17 - (-9) + 1 = 27\). d. Find \(N \cap M'\): - \(M' = \{-20,16,17,18,19,20\}\). - \(N = \{-9,-8,\ldots,10\}\). - Intersection is empty since no elements of \(M'\) are in \(N\). 4. **Final answers:** - a. \(M' = \{-20,16,17,18,19,20\}\) - b. \(N \cap P = \{9,10\}\) - c. \(n(N \cup P) = 27\) - d. \(N \cap M' = \emptyset\)