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Reflexive Closure

Definition: Let R be a binary relation on S. The reflexive closure of R on S is the smallest relation that contains R and is reflexive on S:

reflexiveS(R) :=

such R' subset S x S:

R subset R' /\  R' is reflexive on S /\

forall R": (R subset R" /\  R" is reflexive on S) => R' subset R".

Proposition:

forall S, R: R subset S x S => reflexiveS(R) = R union {<x, x>: x in S}.

Add reflexivity to relation.