Show that the relation R in the set A $= \{1, 2, 3, 4, 5\}$ given by $R = \{ (a,\;b):|a - b|\;is\;even\} ,$ is an equivalence relation. Show that all the elements of $\{1, 3, 5\}$ are related to each other and all the elements of $\{2, 4\}$ are related to each other. But no element of $\{1, 3, 5\}$ is related to any element of $\{2, 4\}$.

We have A $= \{1, 2, 3, 4, 5\}$

$R = \{ (a,\;b):|a - b|\;\;is\;\;even\} ,\;a,\;b \in A$

(i) Reflexive
For any a$\in$A, we have $|a - a| = 0,$ which is even
$\Rightarrow$ $(a,\;a) \in R\forall a \in A$
So, R is reflexive..

(ii) Symmetry
Let $a,\;b \in A.$

Let $(a,\;b) \in R,\;\,then\;\;|a - b|\;\,is\;\;even \Rightarrow \;|b - a|\;\;is\;even$

$\Rightarrow$ $(b,\;a) \in R$

Thus, $(a,\;b) \in R \Rightarrow (b,\;a) \in R$

So, R is symmetric.

(iii) Transitive

Let $a,\;b,\;c \in A.$ Let $(a,\;b) \in R\;\;and\;\;(b,\;c) \in R$

$\Rightarrow$ $|a - b|\;\;$ is even and $|b - c|$ is even

$\Rightarrow$ (a and b both are even or both are odd) and (b and c both are even or both are odd)

Case I : 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 [Therefore, b is even]
$\Rightarrow$ $|a - c|$ is even $\Rightarrow$ $(a,\;c) \in R$

Case II : When b is odd

Let $(a,\;b) \in R$ and $(b,\;c) \in R \Rightarrow \;|a - b|$ is even and $|b - c|$ is even

$\Rightarrow$ a is odd and c is odd [ b is odd]

$\Rightarrow$ $|a - c|\;\;is\;even\; \Rightarrow (a,\;c) \in R$

Thus, (a, b)$\in$R and $(b,\;c) \in R \Rightarrow (a,\;c) \in R$

So, R is transitive.

Hence, R is an equivalence relation.

We know that the difference of any two odd (even) natural numbers is always an even natural number.

Therefore, all the elements of set $\{1, 3, 5\}$ are related to each other and all the elements of $\{2, 4\}$ are related to each other.

We know that the difference of an even natural number and an odd natural number is an odd number.

Therefore, no element of $\{1, 3, 5\}$ is related to any element of $\{2, 4\}$.