reflexive closure

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

  1. S is reflexive
  2. R ⊆ S
  3. For any relation S’, if R ⊆ S’ and S’ is reflexive then S ⊆ S’. In other words, S is smallest relation that satisfies 1 and 2.

The reflexive closure S of a binary relation R on a set X can be formally defined as:

S = R ∪ {(x, x) : x ∈ X}

where {(x, x) : x ∈ X} is the identity relation on X. The reflexive closure of a binary relation on a set is the union of the binary relation and the identity relation on the set.

For example, let R be the greater than relation on the set of integers I:

R = {(a, b) | a ∈ I ∧ b ∈ I ∧ a > b}

The reflexive closure is the union of the relation (greater than) and the identity relation (equality) over I. The union of greater than and equality is greater than or equal to, ≥. So the reflexive closure S of the greater than relation R is:

S = {(a, b) | a ∈ I ∧ b ∈ I ∧ a ≥ b}

 

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