Show that the relation R on Z × Z defined by (a, b) R (c, d) if and only if a + d = b + c is an equivalence relation. Note: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

A relation R on $Z \times Z$ defined by $(a, b) R(c, d)$ iff $a+d=b+c.$

we see that,

$a+b=b+a \forall a, b \in z$

Therefore $(a, b) R(a, b) \forall(a, b) \in R$

therefore R is reflexive relation

Let (a, b) R(c, d)

$\begin{aligned} &\Rightarrow a+d=b+c \\ &\Rightarrow c+b=d+a \\ &\Rightarrow(c, d) R(a, b) \end{aligned}$

Therefore R is symmetric relation.

$\begin{aligned} &\text{Let, (a, b) R(c, d) and (c, d) R(a, f)}\\ &\Rightarrow a+d=b+c and c+f=d+e\\ &\Rightarrow(a+d)+(c+f)=(b+c)+(d+e)\\ &\Rightarrow(a+f)+(d+c)=(b+e)+(d+c)\\ &\Rightarrow a+f=b+e\\ &\Rightarrow(a, b) R(e, f)\\ \end{aligned}$

Therefore R is transitive relation.

Therefore R is an equivalence relation.

since, $\quad a \rightarrow d=b+c$

$\Rightarrow d=-a+b+c$

Therefore $[(a, b)]=\{(c,-a+b+c) ; c \in \mathbb{Z}\}$