# Closure of Relations and Equivalence Relations

Closure of Relations :

Consider a relation  on set  may or may not have a property , such as reflexivity, symmetry, or transitivity.

if there is a relation S with property P containing R such that S is the subset of every
relation with property P containing R, then S is called as closure of R with respect to P

We can obtain closures of relations with respect to property  in the following ways –

1. Reflexive Closure –  is the diagonal relation on set . The reflexive closure of relation  on set  is .
2. Symmetric Closure – Let  be a relation on set , and let  be the inverse of . The symmetric closure of relation  on set  is .
3. Transitive Closure – Let  be a relation on set . The connectivity relation is defined as – . The transitive closure of  is .

Example – Let  be a relation on set  with . Find the reflexive, symmetric, and transitive closure of R.

Solution –
For the given set, . So the reflexive closure of  is

For the symmetric closure we need the inverse of , which is
.
The symmetric closure of  is-

For the transitive closure, we need to find .
we need to find  until . We stop when this condition is achieved since finding higher powers of  would be the same.

Since,  we stop the process.
Transitive closure,  –