Which set is not closed under subtraction?

Which set is not closed under subtraction?

• A. Natural numbers
• B. Integers
• C. Rational numbers
• D. Real numbers

Answer: (A)

Closure under subtraction is being tested. A set is closed under an operation if performing that operation on any two elements always yields another element of the same set. Natural numbers fail this test because subtracting can produce a number outside the set. For example, 3 minus 5 equals -2, which is not a natural number. So natural numbers are not closed under subtraction. In contrast, subtracting any two integers stays an integer, subtracting any two rational numbers stays a rational number, and subtracting any two real numbers stays a real number. Therefore those sets are closed under subtraction.

Closure under subtraction is being tested. A set is closed under an operation if performing that operation on any two elements always yields another element of the same set. Natural numbers fail this test because subtracting can produce a number outside the set. For example, 3 minus 5 equals -2, which is not a natural number. So natural numbers are not closed under subtraction.

In contrast, subtracting any two integers stays an integer, subtracting any two rational numbers stays a rational number, and subtracting any two real numbers stays a real number. Therefore those sets are closed under subtraction.