Show that each of the relation R in the set $A = \{ x \in Z:0 \le x \le 12\} ,$ given by

(i) $R = \{ (a,\;b):|a - b|$ is a multiple of 4$\}$

(ii) $R = \{ (a,\;b):a = b\}$

is an equivalence relation.

Find the set of all elements related to 1 in each case.

(i) $A = \{ x \in Z:0 \le x \le 12\}$

Therefore, $A = \{ 0,\;1,\;2,\;3,\;...,\;12\}$
We have, $R = \{ (a,\;b):|a - b|\;\;is\;multiple\;of\;4\}$

(a) Reflexive

For any $a \in A,\;|a - a| = 0$ is a multiple of 4
Thus, $(a,\;a) \in R\;\;\;\therefore \;$ R is reflexive.

(b) Symmetry

For any a, b$\in$A, let (a, b)$\in$R

$\Rightarrow$ $|a - b|\;\;is\;\;multiple\;\;of\;\;4\;\;\; \Rightarrow \;|b - a|\;\,is\;\,multiple\;\,of\;\,4$

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

i.e., $(a,\;b) \in R\;\; \Rightarrow \;(b,\;a) \in R$

Therefore, R is symmetric.

(c) Transitive
For any a, b,c$\in$A, let $(a,\;b) \in R$ and ( b, c)$\in$R

$\Rightarrow$ $|a - b|\;is\;\;mutiple\;\;of\;\;4\;\;and\;\;|b - c|\;\;is\;\;multiple\;\;of\;\;4$

$\Rightarrow$ $|a - c|\; = \;|a - b + b - c|$

$\Rightarrow$ $|a - c|\; = \;|4{k_1} + 4{k_2}|\;\;where\;\;a - b = 4{k_1}\;\;and\;\;b - c = 4{k_2}$

$\Rightarrow$ $|a - c|\; = \;4|{k_1} + {k_2}|$

$\Rightarrow$ $|a - c|$ is multiple of 4

$\Rightarrow$ $(a - c) \in R$

Therefore, R is transitive.

Hence, R is an equivalence relation.

(ii) R $= \{(a, b) : a$ = $b\}$

$\Rightarrow$ $R = \{ (0,\;0),\;(1,\;1),\;.......,\;(12,\;12)\}$ and $A = (0,\;1,\;2,\;............,\;12\}$

(a) Reflexive
$a \in A\;\; \Rightarrow \;a = a\;\, \Rightarrow \;(a,\;a) \in R\;\, \Rightarrow R$ is reflexive.

(b) Symmetry
a, b$\in$A
Let $(a,\;b) \in R \Rightarrow a = b \Rightarrow b = a \Rightarrow (b,\;a) \in R$

$\Rightarrow$ R is symmetric.

(c) Transitive

$a,\;b,\;c \in A,\;\;\;Let\;(a,\;b) \in R\;\; \Rightarrow a = b$
$(b,\;c) \in R\;\; \Rightarrow b = c \Rightarrow a = c \Rightarrow (a,\;c) \in R$
$\Rightarrow$ R is transitive.

Hence, R is an equivalence relation.

Now set of all elements related to 1 in each case.

(i) Required set $= \{(5, 1), (1, 5), (9, 1), (1, 9)\}$

(ii) Required set $= \{(1, 1)\}$