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Reflexive closure

Having discussed properties such as symmetry, anti-symmetry, transitivity and reflexivity, we now demonstrate how it is possible to transform a relation with one property into a relation with another property. This process is known as "closure." "To transform a relation R into a relation with a given property P, we perform the P closure of R. For example, to transform a relation R into one that is reflexive, we perofrm the reflexive closure of R. Roughly speaking, a certain way of closing a relation is a certain way of expanding or extending the relation"(Steinhart, p. 29).

If we want to turn a relation into an equivalence relation, we must make it transitive, reflexive and symmetric. The reason for this, as we just discussed, is because equivalence relations are transitive, reflexive and symmetric. In order to make a relation reflexive, we simply add the pairs of the form (x, x) for any x in X (Steinhart, p. 29). But the relation that results from such an operation is not merely an ordinary reflexive relation, but a special kind of reflexive relation known as reflexive closure (Steinhart, p. 29). It is represented thus:

R = R ∪ { (x, x) | x ∈ X}

Thus, suppose the relation "is-happier-than" obtains among: {(Felipe, Junior), (Junior, Sunny)}. If we add the property of reflexivity to this relation, the relation becomes "is-happier-than-or-as-happy-as." This reflexive closure is represented {(Junior, Felipe), (Felipe, Sunny), (Junior, Junior), (Felipe, Felipe), (Sunny, Sunny)}.

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.