Given a relation $R \subseteq B \times B $, show the existence and uniqueness of the closure of R under the properties that define an equivalence relation. Does the order in which the closures of these properties is produced matter?
My initial attempt:
Let us generate a candidate $S$ by obtaining the transitive closure of the symmetric closure of the reflexive closure. By applying the definition of symmetric closure on the reflexive closure we have that: $A = (Id_{B} \cup R) \cup (Id_{B} \cup R)^{-1}$. By the property of the inverse relation of the union of relations, we have that $(Id_{B} \cup R) \cup (Id_{B} \cup R)^{-1} = (Id_{B} \cup R) \cup (Id_{B}^{-1} \cup R^{-1}) = Id_{B} \cup R \cup R^{-1} $. Now, applying the definition of transitive closure onto A, we have that $$S = \bigcup_{k \in \mathbb{N^{+}} } (Id_{B} \cup R \cup R^{-1})^{k} $$. How do I demonstrate that this relation is indeed the equivalence closure, in the sense that it exists and is unique?