Define $a \sim b$ if $a - b$ is an integer in $\Bbb R$. Show that ${}\sim{} $ is an equivalence relation. Show the classes of equivalence as well.

Question

Define $a \sim b$ if $a - b$ is an integer in $\Bbb R$. Show that ${}\sim{} $ is an equivalence relation. Show the classes of equivalence as well.

Here's my work. Am I correct? I also do not understand how to find equivalence classes.

Reflexive: $ a\sim a$ -> a - a; so ${}\sim{} $ is reflexive.

Symmetric: $a \sim b -> a - b$ then $b \sim a -> b - a$; so ${}\sim{} $ is symmetric.

Transitive: $a \sim b -> a - b$ and $b \sim c -> b - c$ then $a - c$ so $a \sim c$; so ${}\sim{} $ is transitive.

Answer 1

For the equivalence classes, each has a unique representative in $[0,1)$, so the the set of equivalence classes is $$\{x+\mathbf Z\mid 0\le x<1 \}. $$ THis set is usually denoted $\mathbf R/\mathbf Z$ and is called the one-dimensional torus.