Definition:Reflexive Transitive Closure

Let $$\mathcal{R}$$ be a relation on a set $$S$$.

The reflexive transitive closure of $$\mathcal{R}$$ is denoted $$\mathcal{R}^*$$, and is defined as the reflexive closure of the transitive closure of $$\mathcal{R}$$:


 * $$\mathcal{R}^* = \left({\mathcal{R}^+}\right)^=$$

Thus it is the smallest reflexive and transitive relation on $$S$$ which contains $$\mathcal{R}$$.

It can be shown that
 * $$\left({\mathcal{R}^+}\right)^= = \left({\mathcal{R}^=}\right)^+$$.

That is, the transitive closure of the reflexive closure is the same thing as the reflexive closure of the transitive closure.

Thus this operation can also be called the transitive reflexive closure of $$\mathcal{R}$$.