Question

which of the following sets is closed under division? irrational numbers natural numbers non - zero integers non - zero rational numbers

Explanation:

A set is closed under division if dividing any two elements of the set results in another element of the set.

1. Irrational numbers: Dividing two irrationals (e.g., $\frac{\sqrt{8}}{\sqrt{2}} = 2$) can yield a rational number, so they are not closed.
2. Natural numbers: Dividing two natural numbers (e.g., $\frac{1}{2}$) can yield a non-natural number, so they are not closed.
3. Non-zero integers: Dividing two non-zero integers (e.g., $\frac{1}{2}$) can yield a non-integer, so they are not closed.
4. Non-zero rational numbers: A non-zero rational number can be written as $\frac{a}{b}$ where $a,b$ are non-zero integers. Dividing two such numbers: $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}$, where $ad$ and $bc$ are non-zero integers, so the result is also a non-zero rational number. This set is closed under division.

Answer:

non-zero rational numbers