Answered

AProve that for all integers x and $y, { }^{-} x - y = { }^{-} y - x$.


BDoes part (a) imply that subtraction is commutative? Explain.


ANo, the property proven in part (a) is the commutative property of addition. It does not imply that subtraction is commutative.
BYes, the property proven in part (a) is the commutative property of subtraction.
CNo, the property proven in part (a) is the identity property of addition. It does not imply that subtraction is commutative.
DNo, the property proven in part (a) is the associative property of addition. It does not imply that subtraction is commutative.

Verified Answer :

  • Rewrite \(-x - y = -y - x\) as \((-x) + (-y) = (-y) + (-x)\), which identifies the Commutative Property of Addition.
  • Recognize that subtraction is defined as \(x - y = x + (-y)\), so part (a) relies on the commutativity of addition, not subtraction.
  • Conclude that subtraction is not commutative (\(x - y eq y - x\)), as demonstrated by \(5 - 3 eq 3 - 5\).
  • Final answers are: Part (a) is C. Commutative, and Part (b) is A. No, the property proven in part (a) is the commutative property of addition. It does not imply that subtraction is commutative.

Explanation

  1. Proving the Commutative Property of Addition for Negatives

We are tasked with proving that for all integers \(x\) and \(y\), the equation \(-x - y = -y - x\) holds. Subtraction of an integer can be viewed as the addition of its negative. Thus, we can rewrite the expression \(-x - y\) as \((-x) + (-y)\) and the expression \(-y - x\) as \((-y) + (-x)\). The addition of integers is commutative, meaning that for any two integers \(a\) and \(b\), \(a + b = b + a\). Substituting \(a = -x\) and \(b = -y\), we confirm that \((-x) + (-y) = (-y) + (-x)\), which is exactly the Commutative Property of Addition. Therefore, the property demonstrated in part (a) is the Commutative Property.

  1. Analyzing the Commutativity of Subtraction

Next, we address whether the result from part (a) implies that subtraction is commutative. Subtraction is commutative if \(x - y = y - x\) for all integers \(x\) and \(y\). As demonstrated by a simple example, let \(x = 5\) and \(y = 3\). Then \(x - y = 5 - 3 = 2\), while \(y - x = 3 - 5 = -2\). Since \(2 eq -2\), subtraction is not a commutative operation. The result from part (a) specifically confirms that addition is commutative even when using negative numbers. It does not establish that the subtraction operation itself satisfies the commutative property. Thus, the correct conclusion is that the property proven in part (a) is the commutative property of addition, and it does not imply that subtraction is commutative.

Examples

Understanding the properties of operations is fundamental in mathematics. For example, while addition and multiplication are commutative (meaning \(a+b=b+a\) and \(a \times b = b \times a\)), subtraction and division are not. Knowing this helps in simplifying algebraic expressions and solving equations correctly. For instance, in budgeting, if you have \(\$50\) and subtract \(\$20\) (\(50-20=30\)), it is quite different from having \(\$20\) and subtracting \(\$50\) (\(20-50=-30\)). This difference highlights why subtraction does not follow the commutative property.

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