Show that the relation $R$ defined on the set $A=(1,2,3,4,5)$, given by
$R=\{(a, b):|a-b|$ is even $\}$ is an equivalence relation.
In order to show $R$ is an equivalence relation we need to show $R$ is Reflexive, Symmetric and Transitive.
Given that, $\forall a, b \in A, R=\{(a, b):|a-b|$ is even $\}$.
Now,
$\underline{R}$ is Reflexive if $(a, a) \in \underline{R} \underline{\forall} \underline{a} \in \underline{A}$
For any a $\in A$, we have
$|a-a|=0$, which is even.
$\Rightarrow(a, a) \in R$
Thus, $R$ is reflexive.
$\underline{R}$ is Symmetric if $(a, b) \in \underline{R} \Rightarrow \underline{(b, a)} \in \underline{R} \underline{\forall} \underline{a, b} \in \underline{A}$
$(a, b) \in R$
$\Rightarrow|\mathrm{a}-\mathrm{b}|$ is even.
$\Rightarrow|b-a|$ is even.
$\Rightarrow(b, a) \in R$
Thus, $R$ is symmetric.
$\underline{R}$ is Transitive if $(a, b) \in \underline{R}$ and $(b, c) \in \underline{R} \Rightarrow(a, c) \in \underline{R} \forall \underline{a}, b, c \in \underline{A}$
Let $(a, b) \in R$ and $(b, c) \in R \forall a, b, c \in A$
$\Rightarrow|a-b|$ is even and $|b-c|$ is even
$\Rightarrow(\mathrm{a}$ and $\mathrm{b}$ both are even or both odd) and ( $\mathrm{b}$ and $\mathrm{c}$ both are even or both odd)
Now two cases arise:
Case 1 : when b is even
Let $(a, b) \in R$ and $(b, c) \in R$
$\Rightarrow|a-b|$ is even and $|b-c|$ is even
$\Rightarrow a$ is even and $c$ is even $[\because b$ is even $]$
$\Rightarrow|a-c|$ is even [ $\because$ difference of any two even natural numbers is even]
$\Rightarrow(a, c) \in R$
Case $2:$ when b is odd
Let $(a, b) \in R$ and $(b, c) \in R$
$\Rightarrow|\mathrm{a}-\mathrm{b}|$ is even and $|\mathrm{b}-\mathrm{c}|$ is even
$\Rightarrow \mathrm{a}$ is odd and $\mathrm{c}$ is odd $[\because \mathrm{b}$ is odd $]$
$\Rightarrow|\mathrm{a}-\mathrm{c}|$ is even $[\because$ difference of any two odd
natural numbers is even]
$\Rightarrow(a, c) \in R$
Thus, $R$ is transitive on $Z$.
Since $R$ is reflexive, symmetric and transitive it is an equivalence relation on $Z$.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.