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.
