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.

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