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Show that each of the relation $R$ in the set $A=\{x \in Z: 0 \leq x \leq 12\},$ given by $R =\{( a , b ): a = b \}$ is an equivalence relation. Find the set of all elements related to $1$ in each case.
Solution
$R =\{( a , b ): a = b \}$
For any element a $\in A,$ we have $(a,\, a) \in R,$ since $a=a$
$\therefore R$ is reflexive.
Now, let $(a, b) \in R$
$\Rightarrow a=b$
$\Rightarrow b=a \Rightarrow(b, a) \in R$
$\therefore R$ is symmetric.
Now, let $(a, b) \in R$ and $(b, c) \in R$
$\Rightarrow a=b$ and $b=c$
$\Rightarrow a=c$
$\Rightarrow(a,\, c) \in R$
$\therefore R$ is transitive.
Hence, $R$ is an equivalence relation.
The elements in $R$ that are related to $1$ will be those elements from set $A$ which are equal to $1$
Hence, the set of elements related to $1$ is $\{1\}$.
