Reflective shell

The reflexive envelope of a two-place relation on a set is the smallest reflexive relation on that contains. ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle R}$

Mathematical definition

The reflexive envelope of a two-place relation on a set is given by ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle M}$

${\displaystyle S=R\cup \Delta _{M}=R\cup \{(m,m)|m\in M\},}$

wherein the diagonal on call. ${\displaystyle \Delta _{M}}$ ${\displaystyle M}$

example

The reflexive envelope of the <-relation on (more generally on an ordered set ) is the -relation. ${\displaystyle \mathbb {R} }$ ${\displaystyle \leq }$

See also

• Reflexive-transitive envelope

Individual evidence

1. ↑ Werner Nehrlich: Discrete Mathematics. Basic knowledge for computer scientists. A Mathematica-based representation. Fachbuchverlag Leipzig in Carl Hanser Verlag, Munich / Vienna 2003, ISBN 3-446-22300-2 , p. 164.