How do you find the intersection of the rational numbers, and in interval of irrational numbers?

Question

Take for example $Q ∩ [ - \sqrt(2) , \sqrt (2)]$? Would it be $[ - \sqrt(2) , \sqrt (2)]$ or is this untrue since they are not in $Q$?

Answer 1

Of course, it cannot be $[-\sqrt{2}, \sqrt{2}]$ as the endpoints are not rationals. Your question is not exactly clear, the set you have mentioned is closed in the rationals, that's all I can say for now.

Answer 2

It would usually be written $$\mathbb{Q} \cap [-\sqrt{2}, \sqrt{2}]$$ That is equal to $$\mathbb{Q} \cap (-\sqrt{2}, \sqrt{2})$$ because $\sqrt{2} \not \in \mathbb{Q}$.

Note that this is not $[-\sqrt{2}, \sqrt{2}]$ because (for example) $\frac{\sqrt{2}}{2}$ is in the interval $(-\sqrt{2}, \sqrt{2})$ but is not rational.

The set can also be written $$\{ x \in \mathbb{Q} \mid x^2 < 2\}$$ if you want to stay entirely within the rationals.