Void Relation

Void relation : Let A be a set, then $\Phi \subseteq $ A X A and so it is a relation on A. This relation is called the void-relation or empty relation on set A.
In other words, a relation R on set A is called empty relation, if no elements of A is related to any element of A.
For example : The relation R on the set A = {1,2,3,4} defined by
R = {(a,b):a + b = 10}
We observe that a + b $\neq $ 10 for any two elements of set A . Therefore,
(a,b) $\notin $ R for any a, b $\in$ A.
⇒ R does not contain any element A X A
⇒ R is an empty set.
⇒ R is the void relation on A.

Examples on Void Relation

Example 1 : Let A = {1, 2, 5, 8}, then R = {(x, y), x, y $\in$ A, x * y = 7}
Check whether this relation is empty relation or not.
Solution : This is an empty relation in A, since no ordered pair (x, y) A x A $\in$ satisfies x * y = 7.
Example 2 : Let B = {1, 2, 5, 8}, then R = {(x, y), x, y $\in$ A, x - y = 1}
Check whether this relation is empty relation or not.
Solution : This is not an empty relation in B, since ordered pair (2, 1) satisfies x - y = 1.


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