Define a relation on the set of all real numbers $x, y \in \mathbb{R}$:
$x ≃ y$ if and only if $x − y \in \mathbb{Z}$
Prove that this is an equivalence relation, and find the equivalence class of the number $1/3$.
I proved that the relation is:
reflexive
$$x-x = 0 \in \mathbb{Z}$$
symmetric
$$x-y = y-x \in \mathbb{Z}$$
transitive
$$x-y \in \mathbb{Z}$$
$$y-z \in \mathbb{Z}$$
$$x-z \in \mathbb{Z}$$
Are these proofs enough?
I'm stuck on the step where I need to find the equivalence class.
Best Answer
Opps, carefull: $$x-y\ne y-x$$
You should write: if $x\sim y$ then $x-y\in Z$ so $-(x-y) = y-x\in Z$ so $y\sim x$.
Also, if $x\sim y$ and $y\sim z$ then $x-y,y-z\in Z$, so $(x-y)+(y-z) \in Z$ so $x-z\in Z$ so $x\sim z$.
And equivance class is $$Z+{1\over 3} = \{...-{5\over 3},-{2\over 3},{1\over 3}, {4\over 3},{7\over 3},...\}$$