1. **Problem statement:** We have 100 people surveyed with activities: jogging (J), swimming (S), and cycling (C). Given data: - $|J|=50$, $|S|=30$, $|C|=35$ - $|J \cap S|=14$, $|S \cap C|=7$, $|J \cap C|=9$ - $|J \cap S \cap C|=3$ We need to find: a. Number who jog but do not swim or cycle. b. Number who take part in only one activity. c. Number who do not take part in any activity. 2. **Formulas and rules:** - Use inclusion-exclusion principle. - Number in only one activity $= |J| - |J \cap S| - |J \cap C| + |J \cap S \cap C|$ for jogging only, similarly for others. - Number not in any activity $= $ total surveyed $- |J \cup S \cup C|$ - $|J \cup S \cup C| = |J| + |S| + |C| - |J \cap S| - |S \cap C| - |J \cap C| + |J \cap S \cap C|$ 3. **Calculations:** - Number who jog but do not swim or cycle: $$|J \text{ only}| = |J| - |J \cap S| - |J \cap C| + |J \cap S \cap C| = 50 - 14 - 9 + 3 = 30$$ - Number who take part in only one activity: $$|J \text{ only}| = 30$$ $$|S \text{ only}| = |S| - |J \cap S| - |S \cap C| + |J \cap S \cap C| = 30 - 14 - 7 + 3 = 12$$ $$|C \text{ only}| = |C| - |J \cap C| - |S \cap C| + |J \cap S \cap C| = 35 - 9 - 7 + 3 = 22$$ Sum of only one activity: $$30 + 12 + 22 = 64$$ - Number who do not take part in any activity: $$|J \cup S \cup C| = 50 + 30 + 35 - 14 - 7 - 9 + 3 = 88$$ So, $$\text{No activity} = 100 - 88 = 12$$ 4. **Final answers:** a. 30 people jog but do not swim or cycle. b. 64 people take part in only one activity. c. 12 people do not take part in any of these activities.