1. Problem 3: List all elements in $A = \{x \in \mathbb{Z}^+ : (x^2 - 4) < 10\}$. Step 1: Write the inequality: $$x^2 - 4 < 10$$ Step 2: Add 4 to both sides: $$x^2 < 14$$ Step 3: Since $x \in \mathbb{Z}^+$ (positive integers), find all positive integers whose square is less than 14: $$x^2 < 14 \implies x < \sqrt{14} \approx 3.74$$ Step 4: Positive integers less than 3.74 are $1, 2, 3$. Step 5: Check each: - $1^2 - 4 = 1 - 4 = -3 < 10$ ✓ - $2^2 - 4 = 4 - 4 = 0 < 10$ ✓ - $3^2 - 4 = 9 - 4 = 5 < 10$ ✓ Answer: $A = \{1, 2, 3\}$ 2. Problem 4: List all elements in $B = \{x \in \mathbb{N} : x^3$ is an odd number less than 150$\}$. Step 1: Understand that $x^3$ must be odd and less than 150. Step 2: For $x^3$ to be odd, $x$ must be odd. Step 3: Find all odd natural numbers $x$ such that $x^3 < 150$. Step 4: Test odd numbers: - $1^3 = 1 < 150$ ✓ - $3^3 = 27 < 150$ ✓ - $5^3 = 125 < 150$ ✓ - $7^3 = 343 > 150$ ✗ Step 5: So $x = 1, 3, 5$. Answer: $B = \{1, 3, 5\}$ 3. Problem 5: List all elements in $A = \{x \in \mathbb{Z}^+ : (x^2 + 4)$ is an even number less than 200$\}$. Step 1: Write the conditions: - $x^2 + 4$ is even - $x^2 + 4 < 200$ Step 2: Since 4 is even, $x^2$ must be even for the sum to be even. Step 3: $x^2$ is even only if $x$ is even. Step 4: Find all positive even integers $x$ such that: $$x^2 + 4 < 200 \implies x^2 < 196$$ Step 5: $x < \sqrt{196} = 14$ Step 6: Even positive integers less than 14 are $2, 4, 6, 8, 10, 12$ Step 7: Check each: - $2^2 + 4 = 4 + 4 = 8$ even and < 200 ✓ - $4^2 + 4 = 16 + 4 = 20$ even and < 200 ✓ - $6^2 + 4 = 36 + 4 = 40$ even and < 200 ✓ - $8^2 + 4 = 64 + 4 = 68$ even and < 200 ✓ - $10^2 + 4 = 100 + 4 = 104$ even and < 200 ✓ - $12^2 + 4 = 144 + 4 = 148$ even and < 200 ✓ Answer: $A = \{2, 4, 6, 8, 10, 12\}$