what are integers closed under the operation of subtraction while the whole numbers are no
The Correct Answer and Explanation is:
The concept of closure under an operation refers to whether performing that operation on elements within a set always results in an element that is also within the same set.
Integers and Subtraction
Integers are the set of whole numbers and their negatives, which include {…, -3, -2, -1, 0, 1, 2, 3, …}. Integers are closed under subtraction, meaning that when you subtract any two integers, the result will always be another integer. For example:
- 5−3=25 – 3 = 25−3=2, and both 5 and 3 are integers, as well as the result, 2.
- −2−4=−6-2 – 4 = -6−2−4=−6, and both -2 and 4 are integers, as well as the result, -6.
In these examples, performing subtraction on integers results in another integer, which shows that integers are closed under subtraction.
Whole Numbers and Subtraction
Whole numbers, on the other hand, are the set of non-negative integers: {0, 1, 2, 3, 4, …}. Whole numbers are not closed under subtraction. This is because subtracting one whole number from another can result in a negative number, which is not a whole number. For example:
- 3−5=−23 – 5 = -23−5=−2, which is not a whole number since it is negative.
- 0−1=−10 – 1 = -10−1=−1, which is also not a whole number.
Therefore, subtracting whole numbers does not always yield a whole number, making the set of whole numbers not closed under subtraction.
Key Differences in Closure
- Integers: When subtracting any two integers, the result is always an integer, so integers are closed under subtraction.
- Whole numbers: When subtracting larger numbers from smaller ones, the result can be negative, which is not a whole number. Hence, whole numbers are not closed under subtraction.
This difference illustrates why the set of integers is closed under subtraction, while the set of whole numbers is not.
