# What are Universal Relation with Example ?

## Solution :

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

In other words, a relation R on a set is called universal relation, if each element of A is related to every element of A.

Example : Consider the relation R on the set A = {1, 2, 3, 4, 5, 6} defined by R = {(a, b) $$\in$$ R : |a – b| $$ge$$ 0}.

We observe that  |a – b| $$\ge$$ 0 for all a, b $$\in$$ A

$$\implies$$  (a, b) $$\in$$ R for all (a, b) $$\in$$ A $$\times$$ A

$$\implies$$  Each element of set A is related to every element of set A

$$\implies$$  R = A $$\times$$ A

$$\implies$$  R is universal relation on set A

Note : It is to note here that the void relation and relation and the universal relation on a set A are respectively the smallest and the largest relations on set A. Both the empty (or void) relation and the universal relation are sometimes called trivial relations.