1. **Problem 1:** Given sets: $$\varepsilon = \{(x,y) : x,y \text{ whole numbers}, -2 \leq x < 3, -3 \leq y \leq 1\}$$ $$A = \{(x,y) : x^2 + y^2 = 5, x > y\}$$ $$B = \{(x,y) : xy < 0\}$$ List elements of (a) A, (b) B. **Step 1:** Find all $(x,y)$ with $x^2 + y^2 = 5$ where $x,y$ are whole numbers in the given ranges. Possible $x$ values: $-2,-1,0,1,2$ (since $x<3$ and $x \geq -2$) Possible $y$ values: $-3,-2,-1,0,1$ Check pairs where $x^2 + y^2 = 5$: - $x=\pm 1$, $y=\pm 2$ since $1^2 + 2^2 = 1 + 4 = 5$ - $x=\pm 2$, $y=\pm 1$ since $2^2 + 1^2 = 4 + 1 = 5$ Check $x > y$: - $(1,2)$ no since $1 \not> 2$ - $(1,-2)$ yes since $1 > -2$ - $(-1,2)$ no since $-1 \not> 2$ - $(-1,-2)$ yes since $-1 > -2$ - $(2,1)$ yes since $2 > 1$ - $(2,-1)$ yes since $2 > -1$ - $(-2,1)$ no since $-2 \not> 1$ - $(-2,-1)$ no since $-2 \not> -1$ So, $A = \{(1,-2), (-1,-2), (2,1), (2,-1)\}$. **Step 2:** Find elements of $B = \{(x,y) : xy < 0\}$ with $x,y$ in the given ranges. This means $x$ and $y$ have opposite signs. Possible $x$ values: $-2,-1,0,1,2$ Possible $y$ values: $-3,-2,-1,0,1$ Exclude $x=0$ or $y=0$ since $0 \times$ anything $=0$ not less than 0. Pairs where $x$ and $y$ have opposite signs: - $x>0$, $y<0$: $(1,-3),(1,-2),(1,-1),(2,-3),(2,-2),(2,-1)$ - $x<0$, $y>0$: $(-2,1),(-1,1)$ So, $B = \{(1,-3),(1,-2),(1,-1),(2,-3),(2,-2),(2,-1),(-2,1),(-1,1)\}$. --- 2. **Problem 2:** $$\varepsilon = \{x : 1 \leq x \leq 20\}$$ $$P = \{x : x \text{ multiple of } 2\}$$ $$Q = \{x : x \text{ multiple of } 3\}$$ (a) List elements in $(P \cup Q)'$ (complement of union). (b) Describe elements in $P \cap Q$. **Step 1:** List $P$ and $Q$: - $P = \{2,4,6,8,10,12,14,16,18,20\}$ - $Q = \{3,6,9,12,15,18\}$ **Step 2:** $P \cup Q = \{2,3,4,6,8,9,10,12,14,15,16,18,20\}$ **Step 3:** $(P \cup Q)' = \varepsilon \setminus (P \cup Q) = \{1,5,7,11,13,17,19\}$ **Step 4:** $P \cap Q = \{6,12,18\}$ These are multiples of both 2 and 3, i.e., multiples of 6. --- 3. **Problem 3:** $$\varepsilon = \{x : 1 \leq x \leq 100\}$$ $$S = \{x : x \text{ perfect square}\}$$ $$T = \{x : x \text{ integer having at least one } 6\}$$ $$P = \{x : x \text{ integer ending with } 7\}$$ (a) Find $n(S \cap T)$ (b) Find $n(S \cap P)$ **Step 1:** List perfect squares up to 100: $1,4,9,16,25,36,49,64,81,100$ **Step 2:** Identify which have digit 6 (set $T$): Numbers with digit 6: $6,16,26,36,46,56,60-69,76,86,96$ **Step 3:** $S \cap T$ are perfect squares with digit 6: From perfect squares: $16,36,64$ Count: $n(S \cap T) = 3$ **Step 4:** $P$ numbers ending with 7: $7,17,27,37,47,57,67,77,87,97$ **Step 5:** $S \cap P$ are perfect squares ending with 7: None of the perfect squares end with 7. So, $n(S \cap P) = 0$ --- 4. **Problem 4:** 40 students total. - 23 got distinction in Physics (P) - 15 got distinction in Chemistry (C) - 12 got no distinction in either Find number who got distinction in both Physics and Chemistry. **Step 1:** Number who got distinction in at least one subject: $40 - 12 = 28$ **Step 2:** Use formula: $$n(P \cup C) = n(P) + n(C) - n(P \cap C)$$ $$28 = 23 + 15 - n(P \cap C)$$ **Step 3:** Solve for $n(P \cap C)$: $$n(P \cap C) = 23 + 15 - 28 = 10$$ --- 5. **Problem 5:** $$A = \{x : x \text{ positive integer}, 2^x < 200\}$$ $$B = \{x : x \text{ prime}, x < 10\}$$ List (a) A, (b) B, (c) $A \cap B$. **Step 1:** Find $A$: Check $2^x < 200$ for positive integers $x$: - $2^1=2 < 200$ - $2^2=4 < 200$ - $2^3=8 < 200$ - $2^4=16 < 200$ - $2^5=32 < 200$ - $2^6=64 < 200$ - $2^7=128 < 200$ - $2^8=256 \not< 200$ So, $A = \{1,2,3,4,5,6,7\}$ **Step 2:** Find $B$ (primes less than 10): $2,3,5,7$ **Step 3:** Find $A \cap B$: $\{2,3,5,7\}$ --- **Final answers:** 1. (a) $\{(1,-2), (-1,-2), (2,1), (2,-1)\}$ (b) $\{(1,-3),(1,-2),(1,-1),(2,-3),(2,-2),(2,-1),(-2,1),(-1,1)\}$ 2. (a) $\{1,5,7,11,13,17,19\}$ (b) Multiples of 6 3. (a) 3 (b) 0 4. 10 5. (a) $\{1,2,3,4,5,6,7\}$ (b) $\{2,3,5,7\}$ (c) $\{2,3,5,7\}$