સાબિત કરો કે ગણ $A=\{1,2,3,4,5\}$ પર વ્યાખ્યાયિત સંબંધ $R =\{(a, b):|a-b|$ યુગ્મ છે $\} $ સામ્ય સંબંધ છે. સાબિત કરો કે $\{1,3,5\}$ ના બધા જ ઘટકો એકબીજા સાથે સંબંધ $R$ ધરાવે છે અને $ \{2,4\}$ ના બધા જ ઘટકો એકબીજા સાથે સંબંધ $R$ ધરાવે છે. પરંતુ $\{1,3,5\}$ નો એક પણ ઘટક $ \{2,4\}$ ના કોઈ પણ ઘટક સાથે સંબંધ $R$ ધરાવતો નથી.

$A =\{1,2,3,4,5\}$ and $R =\{( a , b ):| a - b |$ is even $\}$

It is clear that for any element $a \in A$, we have $|a-a|=0$ (which is even).

$\therefore R$ is reflexive.

Let $(a, b) \in R$

$\Rightarrow|a-b|$ is even

$\Rightarrow|-(a-b)|=|b-a|$ is also even

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

$\therefore R$ is symmetric.

Now, let $(a, b) \in R$ and $(b, c) \in R$

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

$\Rightarrow(a-b)$ is even and $(b-c)$ is even

$\Rightarrow(a-c)=(a-b)+(b-c)$ is even             [Sum of two even integers is even]

$\Rightarrow|a-b|$ is even.

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

$\therefore R$ is transitive.

Hence, $R$ is an equivalence relation.

Now, all elements of the set $\{1,2,3\}$ are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even.

Similarly, all elements of the set $\{2,4\}$ are related to each other as all the elements of this subset are even.

Also, no element of the subset $\{1,3,5\}$ can be related to any element of $\{2,4\}$ as all elements of $\{1,3,5\}$ are odd and all elements of $\{2,4\}$ are even. Thus, the modulus of the difference between the two elements (from each of these two subsets) will not be even $[$ as $1-2,\,1-4$, $3-2,\,3-4$, $5-2$ and $5-4$  all are odd $]$