Show that R is an equivalence relation on set of integers.

Question

x,y into a Z (set of integers) and x related y if and only if x-y is a multiple of 3. Show that R is an equivalence relation on Z (set of integers).

I need to show proof. I don't know how to prove Transitivity. This is what I have done but don't know if is right.

Reflexive: x-x=0. So 0 is a multiple of 3. And 0 is an integer.

Symmetric: if x-y is a multiple of 3 then y-x is a multiple of 3.

Transitive: if x-y is a multiple of 3. Then y-z is a multiple of 3. So this leads to x-z is a multiple of 3.

Just to clarify, you are asking to prove that $\mathcal{R}$ is an equivalence relation with $x \mathcal{R} y$ iff $x - y \equiv 0 \mod 3$? — Edward Evans, Mar 15 '16 at 21:56

score 2 · Answer 1 · answered Mar 15 '16 at 21:55

2

Hint: $(x-y)+(y-z) = x-z$

What do you know about adding multiples of a number $n$?

answered Mar 15 '16 at 21:55

Aurey

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Don't understand what you mean? Can you please explain? – user3023185 Mar 15 '16 at 22:28
You mean the second sentence? Both $(x-y)$ and $(y-z)$ are multiples of three. What does that tell you about their sum? – Aurey Mar 15 '16 at 22:42
I do understand I have used (6-3) + (3-9)=6-9 ((x-y) + (y-z) = x-z) . 3+(-6)=(-3) and x-z = -3 also so yes x-z is a multiple of 3. Am I right? – user3023185 Mar 15 '16 at 23:11
$3n + 3m = 3(n+m)$, $n, m, n+m \in \mathbb{Z}$ – Aurey Mar 15 '16 at 23:17

score 2 · Accepted Answer · answered Mar 15 '16 at 22:45

Let $\mathcal{R}$ be an equivalence relation such that $x\mathcal{R}y$ if and only if $x - y = 3k$ with $x, y, k \in \mathbb{Z}$. You have three things to prove.

(a) Is $\mathcal{R}$ reflexive? Clearly, as you have shown, $x - x = 0$ and $0$ surely is a multiple of $3$.

(b) Is $\mathcal{R}$ symmetric? If $x - y = 3k$ then $y - x = -3k = 3(-k) = 3j$ where $j = -k$.

(c) Is $\mathcal{R}$ transitive? Let $x - y = 3k$ and $y - z = 3j$. Then $(x - y) + (y - z) = 3k + 3j = 3(k + j).$ Now let $l = k+j$. Then $x - y + y - z = x - z = 3l$.

From these we conclude that $\mathcal{R}$ is an equivalence relation.

Does this help?

Wonderful, hopefully you can use this in other such problems! — Edward Evans, Mar 15 '16 at 23:16

Show that R is an equivalence relation on set of integers.

2 Answers2