1. **Problem statement:** A department store surveyed 1603 shoppers. - 709 made a purchase. - 951 were satisfied with the service. - 481 of those who made a purchase were not satisfied. Find: 1. Number who made a purchase and were satisfied. 2. Number who made a purchase or were satisfied. 3. Number who were satisfied but did not make a purchase. 4. Number who were not satisfied and did not make a purchase. --- 2. **Formulas and rules:** - Let $P$ = shoppers who made a purchase. - Let $S$ = shoppers who were satisfied. - Use the formula for union: $$|P \cup S| = |P| + |S| - |P \cap S|$$ - Complement rule: total shoppers = 1603. --- 3. **Step 1: Find $|P \cap S|$ (purchase and satisfied):** Given 709 made a purchase, and 481 of these were not satisfied. So satisfied purchasers: $$|P \cap S| = 709 - 481 = 228$$ --- 4. **Step 2: Find $|P \cup S|$ (purchase or satisfied):** Using union formula: $$|P \cup S| = |P| + |S| - |P \cap S| = 709 + 951 - 228 = 1432$$ --- 5. **Step 3: Find satisfied but did not purchase:** $$|S \setminus P| = |S| - |P \cap S| = 951 - 228 = 723$$ --- 6. **Step 4: Find not satisfied and did not purchase:** - Total shoppers = 1603 - Not satisfied shoppers = total - satisfied = 1603 - 951 = 652 - Not purchase shoppers = total - purchase = 1603 - 709 = 894 We want shoppers who are both not satisfied and did not purchase, i.e., $$|\overline{S} \cap \overline{P}| = \text{total} - |P \cup S| = 1603 - 1432 = 171$$ --- **Final answers:** 1. $228$ 2. $1432$ 3. $723$ 4. $171$