Question

To prove that the relation \(R\) on set \(A\) is reflexive, if and only if the complementary relation is irreflexive.

Short answer

To prove that the relation \(R\) on set \(A\) is reflexive, if and only if the complementary relation is irreflexive.

Step-by-step solution

 Given

Consider the relation \(R\) on set \(A\).

The Concept of reflexive relation

Let the relation\(R\)on set\(A\)that is symmetric and transitive. Then\(R\)is reflexive.

Determine the relation

Consider the relation \(R\) on set \(A\) is reflexive.

For example:

if\(\{ a\} \in A \Rightarrow \{ (a,a)\} \in R\); as per the definition of reflexive.

\( \Rightarrow \{ (a,a)\} \notin \bar R{\rm{ }}\& \{ a\} \in A\)

Which indicates irreflexive.

Hence, proved.