1. **Problem 3:** Given a Venn diagram with universal set $\varepsilon = 350$, $n(C) = 200$ (families owning a car), $n(M) = 120$ (families owning a motor-cycle), and subsets labeled $x$, $y$, $z$, and $k$ as shown. 2. We know: $$n(\varepsilon) = x + y + z + k = 350$$ $$n(C) = x + y = 200$$ $$n(M) = y + z = 120$$ 3. (a) To find the smallest possible value of $k$, maximize $x + y + z$ under the constraints. From $n(C)$ and $n(M)$: $$x = 200 - y$$ $$z = 120 - y$$ Substitute into total: $$x + y + z + k = 350 \implies (200 - y) + y + (120 - y) + k = 350$$ Simplify: $$200 - y + y + 120 - y + k = 350$$ $$320 - y + k = 350$$ $$k = 350 - 320 + y = 30 + y$$ Since $k$ must be smallest, minimize $k$ by minimizing $y$. 4. (b) To find the largest possible value of $y$, note that $x, y, z, k \geq 0$. From $x = 200 - y \geq 0 \implies y \leq 200$ From $z = 120 - y \geq 0 \implies y \leq 120$ So $y \leq 120$. Therefore, largest $y = 120$. 5. (a) Smallest $k$ when $y$ is smallest. Since $y \geq 0$, smallest $y = 0$. Then: $$k = 30 + 0 = 30$$ 6. (c) Given $z = 50$, find $x$. From $z = 120 - y$, so: $$50 = 120 - y \implies y = 70$$ From $x = 200 - y$: $$x = 200 - 70 = 130$$ --- 7. **Problem 4:** Given: $$\varepsilon = \{x : x \text{ is an integer and } \frac{1}{2} < x < 18\}$$ $$A = \{x : x \text{ is a multiple of } 3\}$$ $$B = \{x : 9 \leq 3x - 4 < 32\}$$ 8. (a) List elements of $\varepsilon$, $A$, and $B$. - $\varepsilon$ is integers $x$ such that $0.5 < x < 18$, so: $$\varepsilon = \{1, 2, 3, \ldots, 17\}$$ - $A$ multiples of 3 in $\varepsilon$: $$A = \{3, 6, 9, 12, 15\}$$ - For $B$, solve inequalities: $$9 \leq 3x - 4 < 32$$ Add 4: $$13 \leq 3x < 36$$ Divide by 3: $$\frac{13}{3} \leq x < 12$$ Since $x$ integer, $x \geq 5$ and $x < 12$, so: $$B = \{5, 6, 7, 8, 9, 10, 11\}$$ 9. (b) Find $n(A \cup B')$. First find $B'$ (complement of $B$ in $\varepsilon$): $$B' = \varepsilon \setminus B = \{1, 2, 3, 4, 12, 13, 14, 15, 16, 17\}$$ Then: $$A \cup B' = \{3, 6, 9, 12, 15\} \cup \{1, 2, 3, 4, 12, 13, 14, 15, 16, 17\}$$ Combine and remove duplicates: $$A \cup B' = \{1, 2, 3, 4, 6, 9, 12, 13, 14, 15, 16, 17\}$$ Count elements: $$n(A \cup B') = 12$$