identify the property z(x-y) =zx -zy commutative.associative, distributive?
Distributive Property
Z(x+y)=zx+zy
Distributive
To determine whether the given property z(x-y) = zx - zy is commutative, associative, or distributive, let's analyze each property individually:
1. Commutative Property:
The commutative property states that the order of the operands does not affect the result. In other words, if a binary operation * follows the commutative property, then a * b = b * a.
Now, let's evaluate the given equation: z(x-y) = zx - zy.
To check the commutative property, we would need to swap the operands and see if the equation remains true.
Therefore, if we interchange x and y in the equation, we get: z(y-x) = zy - zx.
As we can observe, the equation z(y-x) = zy - zx is not the same as the original equation; therefore, the given property is not commutative.
2. Associative Property:
The associative property states that the grouping of the operands does not affect the final result. In other words, if a binary operation * follows the associative property, then (a * b) * c = a * (b * c).
To determine if the given property is associative, we need to check two equations:
Equation 1: (z(x-y)) * w
Equation 2: z((x-y) * w)
Expanding the equations, we get:
Equation 1: (zx - zy) * w
Equation 2: z(x - yw)
Comparing the two, we can observe that (zx - zy) * w is not equal to z(x - yw). Hence, the given property is not associative.
3. Distributive Property:
The distributive property states that the multiplication of an element by the sum or difference of two other elements can be expressed as the sum or difference of the individual multiplications. In other words, if a, b, and c are elements, then a * (b + c) = a * b + a * c.
To check whether the given property is distributive, we'll evaluate the equation:
z(x - y) = zx - zy
Expanding both sides, we get:
zx - zy = zx - zy
Since both sides of the equation are identical, the given property is indeed distributive.
To summarize:
- The property z(x-y) = zx - zy is not commutative.
- The property z(x-y) = zx - zy is not associative.
- The property z(x-y) = zx - zy is distributive.