In mathematics, certain properties are fundamental to the operations of addition and subtraction. The question here asks for the identification of a property related to the subtraction of numbers and whether the grouping affects the result.
Let's break down the given expression to see if it holds true:
```
(a - b) - c
and
a - (b - c)
```
We need to check whether they are equal for general numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
Consider some numerical examples:
1. Let [tex]\(a = 10\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = 2\)[/tex]:
```
(10 - 5) - 2 = 5 - 2 = 3
10 - (5 - 2) = 10 - 3 = 7
```
Here, [tex]\((10 - 5) - 2 = 3 \neq 7 = 10 - (5 - 2)\)[/tex].
2. Now let [tex]\(a = 20\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 6\)[/tex]:
```
(20 - 8) - 6 = 12 - 6 = 6
20 - (8 - 6) = 20 - 2 = 18
```
Again, [tex]\((20 - 8) - 6 = 6 \neq 18 = 20 - (8 - 6)\)[/tex].
By these examples, we see that altering the grouping of numbers in subtraction does not yield the same result. Therefore, subtraction is not associative.
Given the incorrect options:
- Identity Property of Subtraction: Subtraction does not have an identity property similar to addition.
- Associative Property of Addition: This property states that for addition, the way numbers are grouped does not change the sum. This is not relevant as our operation is subtraction.
- Commutative Property of Subtraction: This property would assert that [tex]\(a - b = b - a\)[/tex], which clearly isn't true for subtraction.
- Inverse Property of Addition: This property states that for every number [tex]\(a\)[/tex], there is a number [tex]\(-a\)[/tex] so that [tex]\(a + (-a) = 0\)[/tex]. This property applies to addition, not to subtraction.
Thus, the best conclusion we can draw is:
The property illustrated is not true, as subtraction does not have the associative property demonstrated by the expectation from the question.
