Let's define relation $ \mathcal R $ in $\Bbb R$ in the following way:
$ x \mathcal R y \iff (x=y \space \lor \space \exists k \in \Bbb Z (x,y \in (k,k+1))$
Describe equivalence classes.
Well, after lot of my effort i can't solve this problem. This is my thoughts
Equivalence class will look as follows (i think):
$[x] = \{ y \in \Bbb R: x \mathcal R y \}$
Im thinking now about truth this sentence : $(x=y \space \lor \space \exists k \in \Bbb Z (x,y \in (k,k+1))$
There can be three possible ways : 1) first sentence true, second also true 2) first false, second true 3) first true, second false
Note 1. $(x=y \space \lor \space \exists k \in \Bbb Z (x,y \in (k,k+1))$ Note 2. $(x\neq y \space \lor \space \exists k \in \Bbb Z (x,y \in (k,k+1))$ Note 3. $(x=y \space \lor \space \forall k \in \Bbb Z (x,y \notin (k,k+1))$
Now i'm stuck. Propably that is not a proper way to analyse this problem and i shouldn't consider three cases here.
Well, if i'll put as class representative integers in situatuion $ x = y $ i'll get something like $[x] = \{ k \}$. When im taking other number from $\Bbb R$ set different than $\Bbb Z$ i have problems, bacause propably i have to use second sentence, i mean: If $\Bbb R-\Bbb Z $ equivalence class will look like $[x] = \{\space \exists k \in \Bbb Z : y \in (k,k+1) \}$ .
Im struggling also with Note 2. (If it's all true, and for example i'll take $x = 0$ and $ y = 100,5 $ i can't find any integer number such that $y \in (k,k+1)$. In note 3 i have similar problems.
Please give me some hints, because i really want to finally unterstand how equivalence relation and equivalence classes works in harder problem (prop it is easy example for u).