Question

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\left\{\right(a,b):|a-b|iseven\}$, is an equivalence relation.

Answer 1

Given $A=\{1,2,3,4,5\}$ and $R=\left\{\right(a,b):|a-b|iseven\}$
To prove that it is equivalent relation we need to prove that $R$ is reflexive, symmetric and transitive.
(i) Reflexive:
Let $aϵA$
then $|a-a|=0$ is an even number
$\therefore (a,a)ϵR,\forall aϵA$
$\therefore R$ is reflexive
(ii) Symmetric
Let $a,bϵA$
$\forall (a,b)ϵR\Rightarrow |a-b|$ is even
$\Rightarrow |-(b-a)|$ is even
$\Rightarrow |b-a|$ is even
$\Rightarrow |b-a|ϵR$
or $(b,a)ϵR$
$\therefore R$ is symmetric
(iii) Transitive
Let $a,b,cϵA$
$\forall (a,b)ϵR$ and $(b,c)ϵR$
we have $|a-b|$ is even and $|b-c|$ is even
$\Rightarrow a-b$ is even and $b-c$ is even
$\Rightarrow a-b$ is even and $b-c$ is even
$\Rightarrow (a-b)+(b-c)$ is even
$\Rightarrow a-c$ is even
$\Rightarrow |a-c|$ is even $\Rightarrow (a,c)ϵR\therefore R$ is transitive
$\therefore R$ is an equivalence relation.