# Why the set of integers {...-3, -2,-1,0, 1, 2, 3..) is NOT "closed" for division?

Dec 7, 2016

When we apply division to the elements of S we get a whole slew of new numbers that are NOT in S, but rather 'outside', so S is not closed with respect to division.

#### Explanation:

For this question, you need a set of numbers (let's say it's called S) and that's all we work with, except we also need an operator, in this case division, that works on any two elements of the set S.

For a set of numbers to be closed for an operation, the numbers and the answer have to belong to that set.

Well, we have a problem because while $5 \mathmr{and} 0$ are both elements of S, $\frac{5}{0}$ is undefined, and so it is not part of S.

Also, $3 \mathmr{and} 4$ are both elements of S, but $\frac{3}{4} \mathmr{and} \frac{4}{3}$ are fractional numbers and so can't be part of S, which is a set of integers.

When we apply division to the elements of S which are all integers, we get a whole slew of new numbers that are NOT in S, but rather 'outside', so S is not closed with respect to division.