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) closure property for division
b) distributive property for division over addition.
a. Closure property for division means that the result of the division of numbers in the given set belongs to the set.
For example, for the set {1,-1}, the closure for division is true, because (-1/1)=-1, and (1/-1)=-1. So all possible divisions of numbers in the set yield a result also in the set.
For the set {1,2}, closure for division is not true. Although 2/1=1 is in the set, 1÷2 = 1/2 is not in the set. So 1÷2 is a counter example of closure.
Try to figure out a few counter examples, and post if you have doubts.
b. Distributive property of multiplication over addition is the following:
5*(2+3)=5*2+5*3=25
Does this work for division?
5/(2+3)=? 5/2 + 5/3 = 25/6 = 4 1/6
The preceding example is a counter example of the distributive property for division over addition.
-9, 3,6
a) To show that the closure property for division is false for the set of integers {-3,-2,-1,0,1,2,3}, we need to find an example where division of two integers does not give us an integer.
Counterexample: Let's consider dividing 3 by 2.
3 ÷ 2 = 1.5
Since 1.5 is not an integer, this counterexample shows that the closure property for division is false for the given set of integers.
b) To show that the distributive property for division over addition is false for the set of integers {-3,-2,-1,0,1,2,3}, we need to find an example where the distributive property does not hold.
Counterexample: Let's consider the expression (3 + 2) ÷ 2.
(3 + 2) ÷ 2 = 5 ÷ 2 = 2.5
On the other hand, if we separately divide each term and then add them, we get:
(3 ÷ 2) + (2 ÷ 2) = 1.5 + 1 = 2.5
Since the results are different, this counterexample shows that the distributive property for division over addition is false for the given set of integers.
To find a counterexample for each of the given generalizations, we need to provide a specific example that disproves the statement. Let's analyze each generalization individually:
a) Closure Property for Division:
The closure property for division states that if you divide any two numbers in a set, the result should still belong to that set.
To show that this is false for the given set of integers {-3, -2, -1, 0, 1, 2, 3}, we need to find a pair of numbers within the set whose division does not result in an integer from the same set.
One such counterexample would be dividing 3 by -2:
3 / -2 = -1.5
As you can see, -1.5 is not an integer, so the division of 3 by -2 does not belong to the set of integers {-3, -2, -1, 0, 1, 2, 3}. This disproves the closure property for division.
b) Distributive Property for Division over Addition:
The distributive property for division over addition states that if you divide a number by the sum of two other numbers, it should be equivalent to dividing the number by each of the two numbers separately and then adding the results.
To show that this is false for the given set of integers {-3, -2, -1, 0, 1, 2, 3}, we need to find a counterexample where the property does not hold.
Suppose we want to disprove the distributive property for division over addition using the numbers -3, 1, and 2. Let's compute both sides of the equation:
Left-hand side:
-3 / (1 + 2) = -3 / 3 = -1
Right-hand side (dividing each term separately and then adding the results):
(-3 / 1) + (-3 / 2) = -3 - 1.5 = -4.5
As you can see, -1 is not equal to -4.5, so the distributive property for division over addition does not hold in this case.
Therefore, we have provided counterexamples to show that both the closure property for division and the distributive property for division over addition are false for the given set of integers.