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