# Definition:Reflexive Transitive Closure

## Definition

Let $\RR$ be a relation on a set $S$.

### Smallest Reflexive Transitive Superset

The **reflexive transitive closure** of $\RR$ is denoted $\RR^*$, and is defined as the smallest reflexive and transitive relation on $S$ which contains $\RR$.

### Reflexive Closure of Transitive Closure

The **reflexive transitive closure** of $\RR$ is denoted $\RR^*$, and is defined as the reflexive closure of the transitive closure of $\RR$:

- $\RR^* = \paren {\RR^+}^=$

### Transitive Closure of Reflexive Closure

The **reflexive transitive closure** of $\RR$ is denoted $\RR^*$, and is defined as the transitive closure of the reflexive closure of $\RR$:

- $\RR^* = \paren {\RR^=}^+$

## Examples

### Arbitrary Example $1$

Let $S = \set {1, 2, 3}$ be a set.

Let $\RR$ be the relation on $S$ defined as:

- $\RR = \set {\tuple {1, 2}, \tuple {2, 2}, \tuple {2, 3} }$

The **reflexive transitive closure** $\RR^*$ of $\RR$ is given by:

- $\RR^* = \set {\tuple {1, 1}, \tuple {1, 2}, \tuple {2, 2}, \tuple {2, 3}, \tuple {1, 3}, \tuple {3, 3} }$

### Arbitrary Example $2$

Let $S = \set {1, 2, 3, 4, 5}$ be a set.

Let $\RR$ be the relation on $S$ defined as:

- $\RR = \set {\tuple {1, 2}, \tuple {2, 3}, \tuple {3, 4}, \tuple {5, 4} }$

The **reflexive transitive closure** $\RR^*$ of $\RR$ is given by:

- $\RR^* = \set {\tuple {1, 2}, \tuple {1, 3}, \tuple {1, 4}, \tuple {2, 3}, \tuple {2, 4}, \tuple {3, 4}, \tuple {5, 4}, \tuple {1, 1}, \tuple {2, 2}, \tuple {3, 3}, \tuple {4, 4}, \tuple {5, 5} }$

## Also see

- Results about
**reflexive transitive closures**can be found**here**.

## Sources

- 1979: John E. Hopcroft and Jeffrey D. Ullman:
*Introduction to Automata Theory, Languages, and Computation*... (previous) ... (next): Chapter $1$: Preliminaries: $1.5$ Relations: Closures of Relations