True, integers are closed under subtraction.
Understanding Closure Under Subtraction
A set of numbers is said to be "closed" under a certain operation if performing that operation on any two numbers within the set always results in another number that is also within the same set. In the case of integers and subtraction, this means that if you subtract any two integers, the result will always be an integer.
Why Integers are Closed Under Subtraction
The set of integers includes all whole numbers (0, 1, 2, 3, ...) and their negative counterparts (..., -3, -2, -1).
- Definition of Subtraction: Subtraction can be thought of as adding the negative of a number. For example, a - b is the same as a + (-b).
- Integers and Negatives: Since the set of integers includes both positive and negative whole numbers, taking the negative of any integer will always result in another integer.
- Closure Under Addition: Integers are closed under addition. That is, adding any two integers always results in an integer.
Since subtraction can be expressed as addition of a negative number, and integers include negative numbers and are closed under addition, integers are therefore closed under subtraction. The reference also confirms this stating: "True, because subtraction of any two integers is always an integer. Therefore, Integers are closed under subtraction." (09-Jan-2020)
Examples
Here are a few examples to illustrate this:
- 5 - 3 = 2 (Both 5, 3, and 2 are integers)
- -2 - 7 = -9 (Both -2, 7, and -9 are integers)
- 10 - (-4) = 10 + 4 = 14 (Both 10, -4, and 14 are integers)
- 0 - 6 = -6 (Both 0, 6, and -6 are integers)
Conclusion
Because subtracting any two integers always produces another integer, integers are closed under subtraction.
