What is this set $\mathbb{R}$ mod $2\pi$ What does it mean for S = $\mathbb{R}$ mod $2\pi$? Can someone please explain as this notation is new to me.
 A: First, define an equivalence relation $\sim$ on $\mathbf{R}$ via $x \sim y$ if there exists an integer $k$, such that $x - y = 2\pi k$. Verify that this is indeed an equivalence relation. Representatives of its equivalence classes are then elements of $\mathbf{R} \mod{2 \pi}$.
Notice that this is analogous to defining modulo integers. 
A: It means that $x\equiv y$ if and only if $x=y+2k\pi$ for some integer $k.$ It is the same when you watch your clock. Then hour $13$ is equal to hour $1$ just because you understand hours mod $12.$ That is $13-1=12.$  
The model for your $\mathbb{R}\: \mathrm{mod}\: 2\pi$ is a circumference of radius $1.$ There you can identify $1$ with $1+2\pi,$ $1-2\pi,$ etc.  
A: $\Bbb{R}\mod M$ means real numbers ($\Bbb{R}$) such that $0\leq x < M$.  So for example $\Bbb{R}\mod  2\pi$ means all real numbers in the half-open interval $[0,2\pi)$, an interval often used when talking about angles or the modulus of a complex number.
Although in set theory that is the end of the story, usually when such a set is used, we are going to also have rules of arithmetic modulo $2\pi$, that is, for example, $3 \cdot \frac{4\pi}{5} = \frac{2\pi}{5}$.
