1. **Problem statement:** We have a class of 70 students playing football, volleyball, and basketball with given intersection counts. We need to answer several questions about the number of students playing specific combinations of these sports. --- 2. **Define variables for each region in the Venn diagram:** - Let $F$ = football, $V$ = volleyball, $B$ = basketball. - Let $x$ = number of students playing only football. - Let $y$ = number of students playing only volleyball. - Let $z$ = number of students playing only basketball. - Given: - Football & Volleyball but not Basketball = 8 - Football & Basketball (including those also playing volleyball) = 21 - Volleyball & Basketball (including those also playing football) = 20 - Football & Volleyball & Basketball = 14 3. **Find the number of students playing exactly two games:** - Exactly two games means the intersection of two sets excluding the third. - Football & Volleyball only = 8 (given) - Football & Basketball only = $21 - 14 = 7$ - Volleyball & Basketball only = $20 - 14 = 6$ - Total playing exactly two games = $8 + 7 + 6 = 21$ 4. **Find the number of students playing only football:** - Total football players = 44 - Football players in intersections: - Football & Volleyball only = 8 - Football & Basketball only = 7 - Football & Volleyball & Basketball = 14 - So, only football = $44 - (8 + 7 + 14) = 44 - 29 = 15$ 5. **Find the number of students playing only volleyball and only basketball:** - Total volleyball players = 42 - Volleyball players in intersections: - Football & Volleyball only = 8 - Volleyball & Basketball only = 6 - Football & Volleyball & Basketball = 14 - Only volleyball = $42 - (8 + 6 + 14) = 42 - 28 = 14$ - Total basketball players = 33 - Basketball players in intersections: - Football & Basketball only = 7 - Volleyball & Basketball only = 6 - Football & Volleyball & Basketball = 14 - Only basketball = $33 - (7 + 6 + 14) = 33 - 27 = 6$ 6. **Find the number of students playing volleyball or basketball but not football:** - Volleyball or basketball but not football = only volleyball + only basketball + volleyball & basketball only - This is $14 + 6 + 6 = 26$ 7. **Find the number of students playing only one game:** - Only one game = only football + only volleyball + only basketball - This is $15 + 14 + 6 = 35$ 8. **Summary answers:** - Only football: $15$ - Volleyball or basketball but not football: $26$ - Exactly two games: $21$ - Only one game: $35$ --- **Exercise 5:** 1. Set $A = \{x \in \mathbb{N}^* \mid |3x + 1| < 10\}$ Solve the inequality: $$|3x + 1| < 10$$ This means: $$-10 < 3x + 1 < 10$$ Subtract 1: $$-11 < 3x < 9$$ Divide by 3: $$-\frac{11}{3} < x < 3$$ Since $x \in \mathbb{N}^*$ (positive integers), possible values are $x = 1, 2$. So, $$A = \{1, 2\}$$ 2. Write sets $B$ and $C$ in comprehension form: - $B = \{1, 2, 3, 4, 6, 8, 12, 24\}$ Observe that $B$ contains divisors of 24. So, $$B = \{x \in \mathbb{N}^* \mid x \text{ divides } 24\}$$ - $C = \{0, 3, 6, 9, 12, 15, 18\}$ Observe that $C$ contains multiples of 3 from 0 to 18. So, $$C = \{x \in \mathbb{N} \mid x = 3k, k \in \mathbb{N}, 0 \leq x \leq 18\}$$