# equivalence relations example

determine if the relation on $\mathbb{Z} \times \mathbb{Z}$ is an equivalence relation.

$$R=\{((a,b),(c,d)) \in A \times A: a+3b=c+3d\}$$

What I have thus far

I need to show that R is reflexive, symmetric and transitive.

reflexive definition: $\forall x \in A ((x,x) \in R)$

symmetric definition: $\forall x \in A \forall y \in A (xRy \rightarrow yRx)$

transitive definition: $\forall x \in A \forall y \in A \forall z \in A((xRy \land yRz) \rightarrow xRz)$

To show R is reflexive

let $x = (a,b)$ then $a+3b = a+3b$, so R is reflexive

To show R is symmetric

If $a+3b = c+3d$ then $c+3d = a+3b$, so R is symmetric

To show R is transitive

let $x = (a,b)$, $y= (c,d)$ and $z = (e,f)$

Suppose $a+3b = c+3d$

and $c+3d = e+3f$

then $a+3b = e+3f$, so R is transitive.

Is this all that needs to be shown?

• Yes, that's all you need to do to show it's an equivalence relation. – Sam Weatherhog Nov 3 '15 at 1:30