Final answer:
The commutative nature of subtraction demonstrated when both x and y are equal to zero is not evidence that subtraction is commutative. Subtraction is not commutative because changing the order of the terms changes the result, unlike addition where A + B always equals B + A regardless of the order.
Step-by-step explanation:
The question asks whether the equation x - y = y - x when x = 0 and y = 0 indicates that subtraction has a commutative property. The correct answer is A. No. An example where x and y both equal 0 and x - y equals y - x does not demonstrate a general property; it simply shows a case where the outputs are the same due to the values of x and y being zero. The commutative property states that two elements can be swapped in an operation without changing the result. This is true for addition, as stated by the property A + B = B + A, which is always true for any numbers A and B.
In contrast, subtraction is not commutative. This can be seen with a simple counterexample. For instance, if we take two different numbers, say 5 and 3, we see that 5 - 3 = 2, but 3 - 5 = -2, which clearly shows that subtraction's outcome depends on the order of the numbers involved. In the case of vectors, while the subtraction may seem commutative because subtracting a vector is like adding its negative (for example, A - B is like A + (-B)), it is important to note that vector addition is commutative (A + B = B + A), but this does not imply that scalar subtraction is commutative.
To understand this further, let's illustrate the subtraction of scalars as addition with a negative: 5 - (+3) equals 5 + (-3) equals 2. When dealing with negative numbers, such as subtracting -6 from 2, we get 2 - (-6) which is 2 + 6 = 8. Here, the minus sign of the number being subtracted has been changed before performing the addition, confirming that subtraction is not implicitly commutative but can be translated to an addition problem with the second term's sign changed.