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.