Take a set $A = [0,\pi]$ and consider the set $B \subset A$ of all numbers $x \in A$ such that $\frac{x}{\pi} = \frac{m}{n}$ where $m,n \in \mathbb{Z}$. Then:
- Is $B$ dense in $A$?
- What is the measure of $B$?
2026-03-25 12:28:06.1774441686
Is the set of numbers $x$ with $\frac{x}{\pi} = \frac{m}{n} $ dense in $[0, \pi]$?
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$B$ is just the set of rational multiples of $\pi$ (which are contained in $A$). It follows immediately that $B$ is dense in $A$ and has measure zero.