Question

Which sets are closed under subtraction? Select all that apply.

a) {..., −3, −2, −1, 0, 1, 2, 3, ...}
b) {1, 2, 3, 4, ...}
c) {1, 3, 5, 7, 9, ...}
d) {rational numbers}

Answer 1

The sets that are closed under subtraction are the set of integers ({..., -3, -2, -1, 0, 1, 2, 3, ...}) and the set of rational numbers, because subtracting any two elements from these sets results in an element that is also within the set.

Step-by-step explanation:

When considering which sets are closed under subtraction, we analyze if performing subtraction on any two elements within the set results in another element that is also within the set.

• Integers ({..., −3, −2, −1, 0, 1, 2, 3, ...}): This set is closed under subtraction because the difference of any two integers is also an integer.
• Natural numbers ({1, 2, 3, 4, ...}): This set is not closed under subtraction. For example, 1 - 2 = -1, which is not a natural number.
• Odd numbers ({1, 3, 5, 7, 9, ...}): This set is not closed under subtraction. For instance, 3 - 5 = -2, which is not an odd number.
• Rational numbers: This set is closed under subtraction, as the difference between any two rational numbers results in another rational number.

Hence, the sets that are closed under subtraction are the set of integers and the set of rational numbers.