Give a counterexample to show that each of the following generalizations about the set of integers {-3,-2,-1,0,1,2,3} is false.
a) commutative property for subtraction
b)associative property for subtraction
To find a counterexample for each generalization, we'll need to provide an example where the property does not hold true within the set of integers {-3, -2, -1, 0, 1, 2, 3}.
a) To show that the commutative property for subtraction is false, we need to find two integers, a and b, where a - b is not the same as b - a.
Let's review the commutative property for subtraction:
a - b = b - a
If this equation is not true for a particular pair of integers, it will serve as a counterexample.
In the set {-3, -2, -1, 0, 1, 2, 3}, let's try a = 2 and b = 3:
2 - 3 = -1
3 - 2 = 1
Since -1 is not equal to 1, this counterexample shows that the commutative property for subtraction does not hold true in this set of integers.
b) Next, let's find a counterexample to disprove the associative property for subtraction.
The associative property for subtraction states that:
(a - b) - c = a - (b - c)
We need to find three integers, a, b, and c, where the left side of the equation is not equal to the right side.
Considering the set {-3, -2, -1, 0, 1, 2, 3}, let's try a = 2, b = 3, and c = 1:
(2 - 3) - 1 = -2 - 1 = -3
2 - (3 - 1) = 2 - 2 = 0
Since -3 is not equal to 0, this counterexample shows that the associative property for subtraction does not hold in this set of integers.
By providing these counterexamples, we have demonstrated that both the commutative and associative properties for subtraction are false within the given set of integers.