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?