## Monday, September 9, 2013

### Boolean algebraic closure

Given a subset of a boolean algebra the closure of the subset under the operations of union, intersection, and complementation is equivalent to a partition of the boolean algebra.

Consider the set system #{#{0 1} #{2 3}} of #{0 1 2 3 4}. The boolean algebraic closure of this set system is defined by partition #{#{0 1} #{2 3} #{4}}. For the set system #{#{0 1} #{0 1 2}} of #{0 1 2 3 4} the resulting partition is #{#{0 1} #{2} #{3 4}}.

I believe this correspondence between boolean subalgebras and equivalence relations demonstrates the close connection between sets and equality. A fundamental aspect of the definition of a set is that a set must not contain elements that are equivalent to one another.