Which statement is true?
A. Odd numbers are closed under addition.
B. Whole numbers are closed under subtraction.
C. Integers are closed under division.
D. Perfect squares are closed under multiplication.
The correct answer and explanation is:
The correct answer is A. Odd numbers are closed under addition.
Explanation:
To understand why this statement is true, we need to define “closed” under a given operation. A set of numbers is said to be closed under an operation (such as addition, subtraction, multiplication, or division) if performing that operation on any two numbers in the set results in a number that is still within the same set.
- Odd numbers are closed under addition: Odd numbers are numbers that have the form 2n + 1, where n is an integer. When you add two odd numbers, the result is always an even number. This means odd numbers are not closed under addition. For example, 3 (odd) + 5 (odd) = 8 (even), which is not odd. Therefore, statement A is false.
- Whole numbers are closed under subtraction: Whole numbers are the set of numbers {0, 1, 2, 3, 4, …}. When you subtract one whole number from another, the result may not always be a whole number. For example, 3 (whole number) – 5 (whole number) = -2, which is not a whole number. Therefore, statement B is false.
- Integers are closed under division: Integers are the set of whole numbers and their negatives. However, division of two integers does not always result in an integer. For example, 1 ÷ 2 = 0.5, which is not an integer. Therefore, statement C is false.
- Perfect squares are closed under multiplication: Perfect squares are numbers that can be written as the square of an integer, like 1, 4, 9, 16, and so on. If you multiply two perfect squares, the result is always another perfect square. For example, 4 (2²) × 9 (3²) = 36 (6²), which is a perfect square. Therefore, statement D is true.
Thus, the correct statement is D. Perfect squares are closed under multiplication.