determine the frontier of the set R\Q (where R is the real numbers and Q is the rational numbers).
I figured R\Q is the same as saying the real line minus all the rational numbers which would just leave the irrational ones. So is the frontier just real line? Sorry I am new to this and I can't understand the reasoning behind this
The elements of $\mathbb{R}\setminus\mathbb{Q}$ are the irrational numbers; the elements of the frontier $\partial(\mathbb{R}\setminus\mathbb{Q})$ (which is the same as $\partial\mathbb{Q}$) are the real numbers $x\in\mathbb{R}$ with the following property (assuming you are considering the standard topology on $\mathbb{R}$): for all $\epsilon>0$, $(x-\epsilon,x+\epsilon)$ intersects both $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$. But since $\mathbb{Q}$ and $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$, any $x\in\mathbb{R}$ has this property. So $\partial(\mathbb{R}\setminus\mathbb{Q})=\mathbb{R}$.