# 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, –  