Show that the relation R on the set

Solution:

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}.