What is Void or Empty Relation with Example ?

Solution :

Let A be a set. Then, \(\phi\) \(\subseteq\) A \(\times\) A and so it is a relation on A. This relation is called the void or empty relation on set A.

In other words, a relation R on a set A is called void or empty relation, if no element of A is related to any element of A.

Example : Consider the relation R on the set A = {1, 2, 3, 4, 5} defined by R = {(a, b) : a – b = 12}.

We observe that a – b \(\ne\) 12 for any two elements of A.

\(\therefore\)   (a, b) \(\notin\) R for any a, b \(\in\) A.

\(\implies\)  R does not contain any element of A \(\times\) A

\(\implies\)  R is empty set

\(\implies\)  R is the void relation on A.

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