Question

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.

Answer 1

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 <i>all possible</i> 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.

Answer 2

-9, 3,6