1. **Problem 1:** In a class of 40 students, 18 are in the Math Club, 15 are in the Science Club, and 7 belong to both clubs. Find how many students are in either club and how many are in neither. Step 1: Identify given values: Total students $N=40$, Math Club $|M|=18$, Science Club $|S|=15$, both clubs $|M \cap S|=7$. Step 2: Calculate the number of students in either club (union) using the formula: $$|M \cup S| = |M| + |S| - |M \cap S| = 18 + 15 - 7 = 26$$ Step 3: Calculate the number of students in neither club (complement): $$\text{Neither} = N - |M \cup S| = 40 - 26 = 14$$ --- 2. **Problem 2:** In a survey of 100 people, 60 like football, 45 like basketball, and 25 like both. Find how many like only football and how many like at least one sport. Step 1: Given: Total $N=100$, Football $|F|=60$, Basketball $|B|=45$, Both sports $|F \cap B|=25$. Step 2: Calculate people who like only football: $$|F| - |F \cap B| = 60 - 25 = 35$$ Step 3: Calculate people who like at least one sport: $$|F \cup B| = |F| + |B| - |F \cap B| = 60 + 45 - 25 = 80$$ --- 3. **Problem 3:** Among 80 surveyed, 50 watched action movies, 30 watched comedy, 20 watched both. Find how many watched only action and how many watched neither genre. Step 1: Given: Total $N=80$, Action $|A|=50$, Comedy $|C|=30$, Both $|A \cap C|=20$. Step 2: Calculate those who watched only action: $$|A| - |A \cap C| = 50 - 20 = 30$$ Step 3: Find number who watched neither: $$|A \cup C| = |A| + |C| - |A \cap C| = 50 + 30 - 20 = 60$$ $$\text{Neither} = N - |A \cup C| = 80 - 60 = 20$$ --- 4. **Problem 4:** Among 60 students, 35 read fiction, 28 read non-fiction, 12 read both. Find how many read only non-fiction and how many read either type. Step 1: Given: Total $N=60$, Fiction $|F|=35$, Non-fiction $|N|=28$, Both $|F \cap N|=12$. Step 2: Find students who read only non-fiction: $$|N| - |F \cap N| = 28 - 12 = 16$$ Step 3: Find students who read either fiction or non-fiction: $$|F \cup N| = |F| + |N| - |F \cap N| = 35 + 28 - 12 = 51$$ --- 5. **Problem 5:** In a survey of 90 people, 40 like pizza, 55 like burgers, and 25 like both. Find how many like only burgers and how many like neither pizza nor burgers. Step 1: Given: Total $N=90$, Pizza $|P|=40$, Burgers $|B|=55$, Both $|P \cap B|=25$. Step 2: Find people who like only burgers: $$|B| - |P \cap B| = 55 - 25 = 30$$ Step 3: Find people who like neither pizza nor burgers: $$|P \cup B| = |P| + |B| - |P \cap B| = 40 + 55 - 25 = 70$$ $$\text{Neither} = N - |P \cup B| = 90 - 70 = 20$$ --- **Final answers:** 1. Either club: 26 students; Neither: 14 students. 2. Only football: 35 people; At least one sport: 80 people. 3. Only action movies: 30 people; Neither genre: 20 people. 4. Only non-fiction: 16 students; Either type: 51 students. 5. Only burgers: 30 people; Neither pizza nor burgers: 20 people.