What is Void or Empty Relation with Example ?

Solution :

Let A be a set. Then, $\phi$ $\subseteq$ A $\times$ A and so it is a relation on A. This relation is called the void or empty relation on set A.

In other words, a relation R on a set A is called void or empty relation, if no element of A is related to any element of A.

Example : Consider the relation R on the set A = {1, 2, 3, 4, 5} defined by R = {(a, b) : a – b = 12}.

We observe that a – b $\ne$ 12 for any two elements of A.

$\therefore$   (a, b) $\notin$ R for any a, b $\in$ A.

$\implies$  R does not contain any element of A $\times$ A

$\implies$  R is empty set

$\implies$  R is the void relation on A.