# Universal Relation

Universal relation is a relation on set A when A X A $\subseteq $ A X A. In other words, universal-relation is the relation if each element of set A is related to every element of A.For example : Relation on the set A = {1,2,3,4,5,6} by

R = {(a,b) $\in$ R : |a -b|$\geq $0}

We observe that |a -b|$\geq $0 for all a, b $\in$ A

$\Rightarrow $(a,b)$\in$ R for all (a,b) $\in$ A X A

$\Rightarrow $ each element of set A is related to every element of set A.

$\Rightarrow $ R = A X A

$\Rightarrow $ R is a universal relation on set A.

**Note :**It is to note here that the void relation and the universal relation on a set A are respectively the smallest and the largest relations on set A.

Both the void and universal relation are sometimes called

**trivial relations**.

## Examples on Universal Relation

**Example : 1**Let A be the set of all students of a boys school. Show that the relation R on A given by R = {(a,b) : difference between the heights of a and b is less than 5 meters} is the universal-relation.

**Solution :**It is obvious that the difference between the heights of any two students of the school has to be less than 5 meters. Therefore (a,b) $\in$ R for all a, b $\in$ A.

$\Rightarrow $ R = A X A

$\Rightarrow $ R is the universal-relation on set A.

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