Reflexive closure - Alchetron, The Free Social Encyclopedia

In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R.

Definition

The reflexive closure S of a relation R on a set X is given by

S = R ∪ { ( x , x ) : x ∈ X }

In words, the reflexive closure of R is the union of R with the identity relation on X.

As an example, if

X = { 1 , 2 , 3 , 4 } R = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ) }

then the relation R is already reflexive by itself, so it doesn't differ from its reflexive closure.

However, if any of the pairs in R was absent, it would be inserted for the reflexive closure. For example, if

X = { 1 , 2 , 3 , 4 } R = { ( 1 , 1 ) , ( 2 , 2 ) , ( 4 , 4 ) }

then reflexive closure is, by the definition of a reflexive closure:

S = R ∪ { ( x , x ) : x ∈ X } = { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) , ( 4 , 4 ) } .

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