Are Whole Numbers Commutative and Associative Under Subtraction?

Understanding Commutativity and Associativity

To understand whether whole numbers are commutative and associative under subtraction, we first need to define these concepts. Commutativity means that the order of the operands does not affect the outcome. Associativity means that the way the operands are grouped does not affect the outcome.

Commutativity of Whole Numbers Under Subtraction

Whole numbers are not commutative under subtraction. This can be shown with a simple counter-example:

Consider the integers 12 and 7. When we subtract 7 from 12, we get:
12 - 7 5

However, if we subtract 12 from 7, we get:

7 - 12 -5

This shows that the order of the operands matters, thus proving that subtraction is not commutative for whole numbers. There are special instances where subtraction might appear commutative, such as 13 - 13 0 and 0 - 0 0, but these are not the general rule.

Associativity of Whole Numbers Under Subtraction

Whole numbers are also not associative under subtraction. To demonstrate this, let's consider a three-term subtraction problem:

Take the expression 20 - 12 - 5. We can group the terms in two different ways:

(20 - 12) - 5 8 - 5 3 20 - (12 - 5) 20 - 7 13

Clearly, the results are different, illustrating that the way the terms are grouped affects the outcome. Therefore, subtraction is not associative for whole numbers.

Subtraction Examples and Counter-Examples

To further illustrate the lack of commutativity and associativity:

Consider the integers 2, 7, and 6. If we evaluate 2 - 7 - 6, we get:

(2 - 7) - 6 -5 - 6 -11

However, if we evaluate 2 - (7 - 6), we get:

2 - 1 1

This shows that the grouping of the terms affects the result, reinforcing the lack of associativity.

Another example is the integers 5 and 3. If we subtract 5 from 3, we get:

3 - 5 -2

But if we subtract 3 from 5, we get:

5 - 3 2

These examples further confirm that subtraction is not commutative.

Finally, consider the integers 3 and 2. If we evaluate 3 - 2, we get:

3 - 2 1

But if we evaluate 2 - 3, we get:

2 - 3 -1

This counter-example clearly demonstrates that whole numbers are not commutative under subtraction.

Conclusion

In summary, whole numbers are neither commutative nor associative under subtraction. Understanding these properties is crucial for grasping the behavior of numbers in various mathematical operations.