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

Course: Foundations of Higher Math

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

Reflexivity:

Let $\displaystyle x \in \mathbb{Z}$.

$\displaystyle 3x-7x=-4x=2(-2x)$ which is even, so *x R x.*

Symmetry:

Let $\displaystyle x,y \in \mathbb{Z}$.

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

$\displaystyle 3x-10x-7y+10y=2a-10x+10y$

$\displaystyle 3y-7x=2a-10x+10y$

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. $\displaystyle \left\{2k|k\in\mathbb{Z}\right\}$

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

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

Quote:

Originally Posted by

**MadSoulz** Course: Foundations of Higher Math

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

Reflexivity:

Let $\displaystyle x \in \mathbb{Z}$.

$\displaystyle 3x-7x=-4x=2(-2x)$ which is even, so *x R x.*

Symmetry:

Let $\displaystyle x,y \in \mathbb{Z}$.

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

$\displaystyle 3x-10x-7y+10y=2a-10x+10y$

$\displaystyle 3y-7x=2a-10x+10y$

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.

Quote:

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. $\displaystyle \left\{2k|k\in\mathbb{Z}\right\}$

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

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.

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

Quote:

Originally Posted by

**HallsofIvy** You still have to prove "transitivity": if x R y and y R z then x R z..

Transitivity:

Let $\displaystyle x,y,z \in \mathbb{Z}$

Assume that *x R y* and *y R z*,

i.e. $\displaystyle 3x-7y=2a$ and $\displaystyle 3y-7z=2b$. $\displaystyle a,b \in \mathbb{Z}$

$\displaystyle 3y-7z=2b$

$\displaystyle \Rightarrow 7y - 7z=2b +4y$

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

Hence *R *is an equivalence relation