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For the equivalence relation on the integers given by $(x, y) ∈ R$ if and only if $7$ divides $x - y$, $7 | (x - y)$. Is $[15]_R = [-13]_R$? Is $[15]_R = [13]_R$ (the $R$'s would be subscripts denoting they are elements of $R$). I'm not sure where to start here, are these asking about symmetry?

Nati0n
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  • This may need some clarification. In your first question, are you asking if 15 is equivalent to -13 in the sense defined earlier? But then, don't you just have to check whether or not 7 divides 15 - (-13) ? – lulu Jul 17 '15 at 02:39
  • Yes it would be in the sense defined earlier, but I guess that's what I'm unsure of if that's what it's asking with that notation – Nati0n Jul 17 '15 at 02:42
  • Well, if it helps at all: it's not a standard notation. At least, I've never seen it before. For a general equivalence relation people just write "x ~ y" or, if you want to emphasize the particular equivalence, maybe "x $\sim_R$ y. In your specific case, by the way, there is a standard notation: $x \equiv y $ mod (7) or simply x = y mod (7). – lulu Jul 17 '15 at 03:02
  • Sorry the notation was just updated, not sure if that helps but I haven't seen it expressed that way before either which is why I ask. – Nati0n Jul 17 '15 at 03:03
  • In any case: I am pretty sure they are just asking you to decide whether or not 15 is equivalent to -13 via the equivalence relation R. – lulu Jul 17 '15 at 03:05
  • Agreed, so would it then be asking if 7 divides 15 - (-13) and 15 - 13 or would it be asking if x mod 7 = 7 mod 7? Sorry for my ignorance on this. – Nati0n Jul 17 '15 at 03:06
  • Let us continue this discussion in chat. – Nati0n Jul 17 '15 at 03:08
  • ok, but only for a short time (nearly bed time in my world). – lulu Jul 17 '15 at 03:20

1 Answers1

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An element $x$ is in the equivalence class ${[y]}_R$ if it satisfies the equivalence relation. $$\begin{align}x\in {[y]}_R \;& \iff\; xRy \\ & \iff\; 7\mid (y-x)\end{align}$$

Now two equivalence classes will be the same if they contain the same elements.   This means that: $${[x]}_R={[y]}_R \;\iff\; xRy$$

So we have $\;{[15]}_R={[{-}13]}_R\;$ iff $\;7\mid({-}13-15)$, and that $\;{[15]}_R={[{+}13]}_R \;$ iff $\;7\mid(13-15)$

Which is true or false?

The truth statement, $\,m\mid n\,$, means $m$ divides $n$ with no remainder. That is, there is some integer $k$ such that $\,km = n\,$ .

Graham Kemp
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