Does $M:=\{x \in \mathbb{Q}: x^2<7\}$ $M\subseteq\mathbb{R}$ have an infimum, supremum, min, max? $M\subseteq\mathbb{R}$
$M:=\{x \in \mathbb{Q}: x^2<7\}$ 
Does $M$ have an infimum, supremum, min, max?
My answer would be that it doesn't because $\sqrt{7}$ and $-\sqrt{7}$ are $\not\in \mathbb{Q}$
Is that correct?

The set does indeed have a supremum and infimum (use the density of $\Bbb Q$ in $\Bbb R$). However, it does not have a max/min as you say, because $M$ has a maximum if and only if $\sup M\in M$ and $M$ has a minimum if and only if $\inf M\in M$.
