The Correct Answer and Explanation is:
(a) Is the set of whole numbers closed under subtraction? Why or why not?
Correct answer:
○ No, for example: 3 − 4 is not positive and is therefore not a whole number.
Explanation:
Whole numbers include 0, 1, 2, 3, and so on — only non-negative integers. A set is closed under an operation if performing that operation on any elements of the set always results in an element that is also in the set. Subtraction is not closed for whole numbers because subtracting a larger number from a smaller one can result in a negative number, which is not a whole number. For example, 3 − 4 = −1, and −1 is not a whole number. Therefore, the set of whole numbers is not closed under subtraction.
(b) Is the set of all integers closed under addition? Why or why not?
Correct answer:
○ Yes, the sum of any two integers is an integer.
Explanation:
Integers include all positive whole numbers, their negatives, and zero. A set is closed under addition if the sum of any two elements in the set is also in the set. For integers, adding any two numbers — whether both are positive, both are negative, or one is negative and one is positive — will always result in another integer. For example:
- (−2) + (3) = 1
- 4 + 5 = 9
- (−3) + (−6) = −9
In each case, the result is an integer. So the set of integers is closed under addition.
(c) Is the set of all integers closed under subtraction? Why or why not?
Correct answer:
○ Yes, if you subtract any integer from another integer, the difference is another integer.
Explanation:
Subtraction of integers also always results in an integer. This is because integers are closed under subtraction — meaning the result of subtracting any integer from another remains within the set of integers. For instance:
- 5 − 3 = 2 (an integer)
- (−3) − (−5) = 2 (an integer)
- 0 − 4 = −4 (an integer)
All of these results remain in the set of integers. Therefore, the set of integers is closed under subtraction.
