Question

Do we necessarily get an equivalence relation when we form the symmetric closure of the reflexive closure of the transitive closure of a relation?

Short answer

Step 1: Form \(R\), the reflexive closure of \(R\) ……(1)

Step 2: Form \(S\), the symmetric closure of \(R\) ……(2)

Step 3: Form \(T\), the transitive closure of \(S\) ……(3)

End: \(T\) is the required smallest equivalence relation containing \(R\). ……(4)

Step-by-step solution

Given data

\(R\) is any given relation

Concept  used of Equivalence relation 

An equivalence relation is a binary relation that is reflexive, symmetric and transitive.

Show  the equivalence relation

Any equivalence relation containing \(R\) must necessarily be reflexive, symmetric and transitive.

Steps 1,2 and 3 ensure this and the minimality of \(T\).

Thus, \((4)\) is verified and \(T\) is the required smallest equivalence relation containing \(R\).

The steps 1, 2 and 3 constitute the algorithm for producing the smallest equivalence relation containing \(R\).