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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.

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

1 Answer 1

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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.

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  • $\begingroup$ One-dimensional torus aka circle :D $\endgroup$
    – Ennar
    Commented May 6, 2020 at 18:59
  • $\begingroup$ Not quite: it has some more structure than the plain circle – rather the special unitary group $U(1)$. $\endgroup$
    – Bernard
    Commented May 6, 2020 at 19:06
  • $\begingroup$ It's a terminological difference, then. When I say circle, I think of the Lie group. To be honest, this is the first time I hear of this difference. $\endgroup$
    – Ennar
    Commented May 6, 2020 at 19:13
  • $\begingroup$ For me, a circle, without context, is just a geometric object. $\endgroup$
    – Bernard
    Commented May 6, 2020 at 19:14

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