Let R be a relation on the set A of ordered pair of integers defined by

Solution:

We observe the following properties of R.

Reflexivity: Let $(a, b)$ be an arbitrary element of the set A. Then,

$(a, b) \in A$

$\Rightarrow a b=b a$

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

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

Symmetry : Let $(x, y)$ and $(u, v) \in A$ such that $(x, y) R(u, v)$. Then,

$x v=y u$

$\Rightarrow v x=u y$

$\Rightarrow u y=v x$

$\Rightarrow(u, v) R(x, y)$

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

Transitivity: Let $(x, y),(u, v)$ and $(p, q) \in R$ such that $(x, y) R(u, v)$ and $(u, v) R(p, q)$.

$\Rightarrow x v=y u$ and $u q=v p$

Multiplying the corresponding sides, we get

$x v \times u q=y u \times v p$

$\Rightarrow x q=y p$

$\Rightarrow(x, y) R(p, q)$

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

Hence, R is an equivalence relation on A