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 –
- Reflexive Closure – is the diagonal relation on set . The reflexive closure of relation on set is .
- Symmetric Closure – Let be a relation on set , and let be the inverse of . The symmetric closure of relation on set is .
- 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.
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, –