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.

12th grade math


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