This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's how to check the closure property for addition for each pair of numbers: The closure property for addition states that if you add any two numbers from a specific set, the result must also be in that same set. All the numbers provided are rational numbers. The set of rational numbers (Q) is closed under addition, meaning the sum of any two rational numbers is always a rational number. a) Step 1: Identify the numbers and calculate their sum. The numbers are (2)/(3) and 6. Both are rational numbers. (2)/(3) + 6 = (2)/(3) + (18)/(3) = (2+18)/(3) = (20)/(3) Step 2: Check if the sum is a rational number. The sum, (20)/(3), is a rational number. The closure property for addition holds for rational numbers. b) Step 1: Identify the numbers and calculate their sum. The numbers are (1)/(8) and (3)/(4). Both are rational numbers. (1)/(8) + (3)/(4) = (1)/(8) + (3 × 2)/(4 × 2) = (1)/(8) + (6)/(8) = (1+6)/(8) = (7)/(8) Step 2: Check if the sum is a rational number. The sum, (7)/(8), is a rational number. The closure property for addition holds for rational numbers. c) Step 1: Identify the numbers and calculate their sum. The numbers are -(2)/(7) and (3)/(4). Both are rational numbers. -(2)/(7) + (3)/(4) = -(2 × 4)/(7 × 4) + (3 × 7)/(4 × 7) = -(8)/(28) + (21)/(28) = (-8+21)/(28) = (13)/(28) Step 2: Check if the sum is a rational number. The sum, (13)/(28), is a rational number. The closure property for addition holds for rational numbers. d) Step 1: Identify the numbers and calculate their sum. The numbers are -(1)/(7) and -(1)/(3). Both are rational numbers. -(1)/(7) + (-(1)/(3)) = -(1)/(7) - (1)/(3) = -(1 × 3)/(7 × 3) - (1 × 7)/(3 × 7) = -(3)/(21) - (7)/(21) = (-3-7)/(21) = -(10)/(21) Step 2: Check if the sum is a rational number. The sum, -(10)/(21), is a rational number. The closure property for addition holds for rational numbers. In all cases, the sum of the two given rational numbers is also a rational number. Therefore, the closure property for addition holds for all these pairs within the set of rational numbers. a) Sum: (20)/(3). The closure property holds. b) Sum: (7)/(8). The closure property holds. c) Sum: (13)/(28). The closure property holds. d) Sum: -(10)/(21). The closure property holds. Send me the next one 📸