Show that the relation R on the set


Show that the relation $R$ on 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 .


We observe the following properties of R.

Reflexivity: Let a be an arbitrary element of A. Then,

$a \in R$

$\Rightarrow a=a$                    [Since, every element is equal to itself]

$\Rightarrow(a, a) \in R$ for all $a \in A$

So, $R$ is reflexive on $A$.

Symmetry: Let $(a, b) \in R$

$\Rightarrow a b$

$\Rightarrow b=a$

$\Rightarrow(b, a) \in R$ for all $a, b \in A$

So, $R$ is symmetric on $A$.

Transitivity: Let $(a, b)$ and $(b, c) \in R$

$\Rightarrow a=b$ and $b=c$

$\Rightarrow a=b c$

$\Rightarrow a=c$

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

So, $R$ is transitive on $A$.

Hence, R is an equivalence relation on A.

The set of all elements related to 1 is {1}.



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