Commutativity Of Subtraction And Closure Of Division In Integers With Examples

Jun 28, 2025 by ADMIN 79 views

Q4) i) Is the subtraction of integers commutative? Give examples. ii) Is the division of integers closed? Give examples.

Understanding Commutativity in Integer Subtraction and Closure in Integer Division

In the realm of mathematics, particularly within number systems, two fundamental properties often come under scrutiny commutativity and closure. These properties dictate how operations behave with respect to the order of operands and whether the result of an operation remains within the same set. In this article, we will delve into the intricacies of these properties, specifically focusing on integer subtraction and division. We aim to provide a comprehensive understanding of whether integer subtraction is commutative and if integer division exhibits closure. By exploring these concepts with illustrative examples, we will clarify their importance in mathematical operations.

The commutative property is a cornerstone concept in mathematics, primarily concerning binary operations. A binary operation, denoted by

, is considered commutative if, for any two elements 'a' and 'b' in a given set, the order in which the operation is performed does not affect the result. Mathematically, this is expressed as:

a * b = b * a

This property holds true for operations like addition and multiplication within the set of integers. For example, 3 + 5 yields the same result as 5 + 3, and similarly, 2 * 7 equals 7 * 2. However, when we consider subtraction, the scenario changes significantly.

Subtraction is an arithmetic operation that finds the difference between two numbers. Unlike addition and multiplication, subtraction does not adhere to the commutative property. To demonstrate this, let's consider two integers, 'a' and 'b'. If subtraction were commutative, then:

a - b = b - a

would hold true for all integers 'a' and 'b'. However, this is not the case. To illustrate this, let's use a numerical example. Consider the integers 5 and 3. If we subtract 3 from 5, we get:

5 - 3 = 2

Now, if we reverse the order and subtract 5 from 3, we obtain:

3 - 5 = -2

Clearly, 2 is not equal to -2. This simple example demonstrates that the order in which integers are subtracted matters, and therefore, subtraction is not commutative. The difference in sign between the two results highlights the non-commutative nature of subtraction.

To further emphasize this point, let's consider a more general algebraic approach. If we assume that subtraction is commutative, then for any integers 'a' and 'b', we would have:

a - b = b - a

Adding (a + b) to both sides of the equation, we get:

a - b + (a + b) = b - a + (a + b)

Simplifying both sides, we have:

2a = 2b

Dividing both sides by 2, we arrive at:

a = b

This result implies that the only way subtraction can be commutative is if the two integers being subtracted are equal. However, the commutative property should hold true for all integers, not just when they are equal. Therefore, our algebraic analysis further confirms that subtraction of integers is not a commutative operation. The order of the numbers in a subtraction problem affects the outcome, making it a non-commutative operation.

The closure property is another essential concept in mathematics that determines whether an operation performed on elements within a set will result in an element that is also within the same set. In simpler terms, a set is said to be closed under a particular operation if performing that operation on any two elements of the set produces a result that is also a member of the set. For instance, the set of integers is closed under addition, subtraction, and multiplication because adding, subtracting, or multiplying any two integers will always yield another integer. However, the same cannot be said for division.

Division is an arithmetic operation that involves splitting a number into equal parts. When we consider the set of integers, division introduces a unique challenge in terms of closure. To illustrate this, let's consider two integers, 'a' and 'b', where 'b' is not zero (as division by zero is undefined). If the set of integers were closed under division, then:

a / b

would always result in an integer. However, this is not the case. To demonstrate this, let's use a numerical example. Consider the integers 5 and 2. If we divide 5 by 2, we get:

5 / 2 = 2.5

The result, 2.5, is not an integer. It is a decimal number, specifically a rational number, but it does not belong to the set of integers. This example clearly shows that dividing one integer by another does not necessarily produce an integer, and therefore, the set of integers is not closed under division.

To further illustrate this point, let's consider another example. If we divide 10 by 3, we get:

10 / 3 = 3.333...

The result, 3.333..., is a repeating decimal, which is also not an integer. This reinforces the idea that division can often lead to results that fall outside the set of integers. Only in specific cases, such as when one integer is a multiple of the other, will the result be an integer. For example, 12 divided by 3 equals 4, which is an integer. However, these cases are exceptions rather than the rule.

The lack of closure under division has significant implications in mathematics. It means that when performing division within the set of integers, we may encounter numbers that belong to a different set, such as rational numbers. This distinction is crucial in various mathematical contexts, including algebra, number theory, and calculus. Understanding that integers are not closed under division helps us appreciate the broader landscape of number systems and the properties that govern them. In summary, the set of integers is not closed under the operation of division because dividing two integers does not always result in another integer. The result may be a fraction or a decimal, which lies outside the set of integers, highlighting the importance of the closure property in defining the boundaries of mathematical operations within specific number sets.

To solidify our understanding of commutativity in subtraction and closure in division, let's explore some additional examples. These examples will further demonstrate the concepts discussed and highlight the nuances of these operations within the set of integers.

Examples for Subtraction

Consider the integers -4 and 7. Subtracting 7 from -4 gives:
```
-4 - 7 = -11
```
Now, subtracting -4 from 7 gives:
```
7 - (-4) = 7 + 4 = 11
```
Here, -11 ≠ 11, which clearly shows that subtraction is not commutative.
Let's take two negative integers, -2 and -5. Subtracting -5 from -2 gives:
```
-2 - (-5) = -2 + 5 = 3
```
Subtracting -2 from -5 gives:
```
-5 - (-2) = -5 + 2 = -3
```
Again, 3 ≠ -3, reinforcing the non-commutative nature of subtraction.
Consider the case of subtracting 0 from an integer. Let's use the integer 9:
```
9 - 0 = 9
```
Subtracting 9 from 0 gives:
```
0 - 9 = -9
```
Here, 9 ≠ -9, which further illustrates that subtraction is not commutative, even when one of the integers is zero.

Examples for Division

Consider the integers 8 and 3. Dividing 8 by 3 gives:
```
8 / 3 = 2.666...
```
The result is a repeating decimal, not an integer. This demonstrates that the set of integers is not closed under division.
Let's take the integers -10 and 4. Dividing -10 by 4 gives:
```
-10 / 4 = -2.5
```
The result is a decimal, not an integer, further confirming the lack of closure in integer division.
Consider dividing 7 by -2:
```
7 / -2 = -3.5
```
The result is again a decimal, not an integer. This example shows that even with negative integers, division does not necessarily result in an integer.
Let's examine a case where the result is an integer. Dividing 15 by 5 gives:
```
15 / 5 = 3
```
In this specific case, the result is an integer. However, this is an exception rather than the rule. The closure property requires that the operation results in an element within the set for all possible pairs of elements, not just some.
Consider dividing any integer by 1. For example, 10 / 1 = 10, which is an integer. However, this does not change the overall conclusion that division is not closed for integers, as many other divisions will result in non-integers.

These examples collectively illustrate that while subtraction is not commutative, division is not closed within the set of integers. The outcomes of these operations can often lead to numbers that fall outside the integer set, reinforcing the importance of understanding these properties in mathematical operations.

In conclusion, our exploration into the properties of integer subtraction and division has revealed important distinctions. Subtraction of integers is definitively not commutative, as the order in which integers are subtracted significantly impacts the result. This non-commutative nature is evident through numerous examples and algebraic analysis.

On the other hand, division of integers is not closed. This means that dividing one integer by another does not always result in an integer. The result can often be a fraction or a decimal, which falls outside the set of integers. The lack of closure in division highlights the boundaries of the integer set and the importance of considering other number systems, such as rational numbers, when performing division.

Understanding these properties is crucial for mastering mathematical operations and ensuring accurate calculations. Commutativity and closure are fundamental concepts that govern how operations behave within specific number sets. By recognizing that subtraction is not commutative and division is not closed for integers, we can avoid common errors and gain a deeper appreciation for the nuances of mathematical operations.

In summary, while addition and multiplication are commutative and closed within the set of integers, subtraction and division present different behaviors. Subtraction is not commutative, and division is not closed. These distinctions are essential for a comprehensive understanding of mathematical principles and their application in various contexts.