I have the relation $x\mathsf{R}y$ if $x – y \in \mathbf{Z}$ where $x, y$ are real numbers. I want to describe the equivalence classes of the relation.
I have proven to myself that $x-y$ is an equivalence relation if $x-y \in \mathbf{Z}$. But, how does this set partition out? Since an equivalence class of a set S containing a is defined as: $[a] = \{x \in S : x\mathsf{R}a\}$ isn't this just the set of all real numbers? i.e. doesn't this simply partition out into $\mathbf{R}$?
I guess I'm confused on how the definition for an equivalence class works. I read the questions and answers related to this question before I submitted it. I think, unfortunately, I need a bit more of an explanation for a complete understanding as I am a complete beginner in abstract algebra. Aren't there an infinite number of equivalence classes here?
Thank you in advance, very much appreciated.
Best Answer
$y$ is equivalent to $x$ if $x-y =k$ for some integer $k$, right? That means $x = y + k$ for some integer $k$.
So, the equivalence class of $y$ is the set of elements of the form $y+k$, where $k$ ranges over the integers, since it is exactly this set that satisfies the above property of $x-y=k$.
So, start with an element $y$. To get its equivalence class, simply shift $y$ by $k$ for each integer $k$, and you have the equivalence class of $y$.
In math, you would write $[y] = \{y +k | k \in \Bbb Z \}$.