Thread: Equivalence relation help

1. Equivalence relation help

Let R be the relation on integers defined by aRb if and only if 2a+3b congruent to 0 (mod 5). Show that this is an equivalence relation.

2. Originally Posted by jconfer
Let R be the relation on integers defined by aRb if and only if 2a+3b congruent to 0 (mod 5). Show that this is an equivalence relation.
Did you start by trying to show this relation has the charcteristics of an equivalence relation?

3. Yeah i got the reflexive one, but i couldn't get transitive or symetry to work.

Thats what i really need help on.

4. Originally Posted by jconfer
Yeah i got the reflexive one, but i couldn't get transitive or symetry to work.

Thats what i really need help on.
ok, symmetry is a bit difficult, i'l admit when seeing this for the first time. so i will do it for you. but i will leave transitivity to you. just express the congruence as i did.

Note that $2a + 3b \equiv 0~\mbox{mod }5$ means $2a + 3b = 5k$ for some $k \in \mathbb{Z}$

Claim: R is symmetric

Proof: Assume $aRb$, that is, $2a + 3b = 5k$ for some integer $k$

Then, $2b + 3a = 5b - 3b + 5a - 2a$

$\Rightarrow 2b + 3a = 5a + 5b - \underbrace{(2a + 3b)}_{5k}$

$\Rightarrow 2b + 3a = 5(a + b - k)$

since $(a + b - k) \in \mathbb{Z}$, we have $bRa$

Thus, R is symmetric