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.

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.

• Welcome to Mathematics Stack Exchange. For reflexive, I would write "$a-a=0$ is an integer, so $a\sim a$" Commented May 6, 2020 at 18:43
• $a-a$ is not a statement, so it makes no sense to say that something implies it; each of your three lines has the same problem.. Commented May 6, 2020 at 18:45
• The equivalence classes are of the form $$[a]_\sim=\{b\mid a-b\in\Bbb Z\}.$$ Commented May 6, 2020 at 18:48
• How would I correct my work for symmetric and transitive? I get that for reflexive a - a = 0 so it can imply a ~ a, but I have no way of knowing what a - b is or b - a is. Commented May 6, 2020 at 19:14
• You have presented no work. What you wrote means nothing. Please read the comment by Brian M. Scott, which was upvoted 7 times. Commented May 6, 2020 at 19:16

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.
• Not quite: it has some more structure than the plain circle – rather the special unitary group $U(1)$. Commented May 6, 2020 at 19:06