I need to find the inf, sup, min, max (if they exist) of this set: $\{ x\in\mathbb{R}\setminus\mathbb{Q}: x<3\}$. I know that there is no inf or min, since the set is not bounded below. I also know that there is no max, since for any $m$ I can always find some other $x\in\mathbb{R}\setminus\mathbb{Q}$ such that $m<x<3$. I want to say that the sup is 3. What is throwing me off is that $3\notin\mathbb{R}\setminus\mathbb{Q}$, and I saw another example talking about a subset of $\mathbb{Q}$ that "didn't have a sup in $\mathbb{Q}$" when it looked like the sup should have been $\sqrt{2}$. So if the question just asks to "find the sup" does that imply I am looking for a sup in all of $\mathbb{R}$ or just in the subset in question?
The other thing that threw me off was that I had to find the inf, sup, min, max (if they exist) of $\{ x\in\mathbb{Q}: x<3\}$ first, and it seemed like I shouldn't get the same answers for both sets.