# Thread: Prove that R is an equivalence relation. Determine the distinct equivalence classes.

1. ## Prove that R is an equivalence relation. Determine the distinct equivalence classes.

Course: Foundations of Higher Math

A relation $R$ is defined on $\mathbb{Z}$ by x R y if $3x-7y$ is even. Prove that $R$ is an equivalence relation. Determine the distinct equivalence classes.

Reflexivity:
Let $x \in \mathbb{Z}$.
$3x-7x=-4x=2(-2x)$ which is even, so x R x.

Symmetry:
Let $x,y \in \mathbb{Z}$.
Assume that x R y ,i.e. $3x-7y=2a$, $a\in \mathbb{Z}$.

$3x-10x-7y+10y=2a-10x+10y$
$3y-7x=2a-10x+10y$
which is even, so y R x.

Is this method fine?

Also, how would I notate the equivalence classes? I believe that there will be 2 distinct classes because 3x and 7y must have same parity for the difference to be even.

1. $\left\{2k|k\in\mathbb{Z}\right\}$
2. $\left\{2k+1|k\in\mathbb{Z}\right\}$?

2. ## Re: Prove that R is an equivalence relation. Determine the distinct equivalence class

Course: Foundations of Higher Math

A relation $R$ is defined on $\mathbb{Z}$ by x R y if $3x-7y$ is even. Prove that $R$ is an equivalence relation. Determine the distinct equivalence classes.

Reflexivity:
Let $x \in \mathbb{Z}$.
$3x-7x=-4x=2(-2x)$ which is even, so x R x.

Symmetry:
Let $x,y \in \mathbb{Z}$.
Assume that x R y ,i.e. $3x-7y=2a$, $a\in \mathbb{Z}$.

$3x-10x-7y+10y=2a-10x+10y$
$3y-7x=2a-10x+10y$
which is even, so y R x.

Is this method fine?
So far but you haven't finished. You still have to prove "transitivity": if x R y and y R z then x R z.

Also, how would I notate the equivalence classes? I believe that there will be 2 distinct classes because 3x and 7y must have same parity for the difference to be even.

1. $\left\{2k|k\in\mathbb{Z}\right\}$
2. $\left\{2k+1|k\in\mathbb{Z}\right\}$?
So you are saying that all odd numbers are equivalent and all even numbers are equivalent?
Yes that is correct and your notation is good.

3. ## Re: Prove that R is an equivalence relation. Determine the distinct equivalence class

Originally Posted by HallsofIvy
You still have to prove "transitivity": if x R y and y R z then x R z..
Transitivity:
Let $x,y,z \in \mathbb{Z}$
Assume that x R y and y R z,
i.e. $3x-7y=2a$ and $3y-7z=2b$. $a,b \in \mathbb{Z}$

$3y-7z=2b$
$\Rightarrow 7y - 7z=2b +4y$
$(3x-7y) + (7y-7z) = 3x-7z = 2a + 2b+4y$ which is even. So, x R z

Hence R is an equivalence relation