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

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

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.

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.

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

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prove that R is an equivalence and determine the equivalence classes of S

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