Let $A = \{ 1,2,3, \ldots ,9\}$ and $R$
be the relation in $A \times A$ defined by
$(a,b)R(c,d)$ if $a + d = b + c$ for
$(a,b),(c,d)$ in $A \times A$. Prove that $R$
is an equivalence relation and also obtain the equivalent class [(2,5)].

It is given that,, $A = \{ 1,2,3, \ldots ,9\}$

and $(a,b)R(c,d)$ if $a + d = b + c$ for $(a,b) \in A \times A$ and $(c,d) \in A \times A$
Let $(a,b)R(a,b)$

$\Rightarrow$ $a + b = b + a,\forall a,b \in A$
which is true for any $a,b \in A$.

Hence we can say that, $R$ is For reflexive.

Let $(a,b)R(c,d)a + d = b + c$

$c + b = d + a \Rightarrow (c,d)R(a,b)$

So, $R$ is For symmetric.
Let $(a,b)R(c,d)$ and $(c,d)R(e,f)$

$a + d = b + c$ and $c + f = d + e$

$a + d = b + c$ and $d + e = c + f$

$(a + d) - (d + e) = (b + c) - (c + f)$

$(a - e) = b - f$

$a + f = b + e$

$(a,b)R(e,f)$

So, $R$ is transitive.

Hence we can say that, $R$ is an equivalence relation.

Now, equivalence class containing [(2,5)] is {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)} .