No, integers are not closed under division.
This means that when you divide one integer by another integer, the result is not always an integer. Closure under an operation requires that performing the operation on elements within the set always produces another element within the same set.
Here's a more detailed explanation:
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Definition of Integers: Integers are whole numbers (not fractions) and their negatives. They include numbers like -3, -2, -1, 0, 1, 2, 3, and so on.
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Closure Property: A set is said to be closed under an operation if performing that operation on any two elements of the set results in another element that is also in the set.
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Division Example: Consider the integers 4 and 9. When we divide 4 by 9 (4 ÷ 9), we get 4/9.
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Why Integers Fail Closure under Division: The result of 4/9 is a fraction. Fractions are not integers. Therefore, since dividing two integers (4 and 9) produced a result (4/9) that is not an integer, integers are not closed under division.
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Examples of Closure: Integers are closed under addition, subtraction, and multiplication because performing these operations on any two integers always results in another integer.
| Operation | Closed Under Integers? | Example | Result |
|---|---|---|---|
| Addition | Yes | 2 + 3 | 5 |
| Subtraction | Yes | 5 - 2 | 3 |
| Multiplication | Yes | 4 * 3 | 12 |
| Division | No | 1 ÷ 2 | 1/2 or 0.5 |
