Editing Algebra of sets (section)

== Algebra of inclusion ==

The following proposition says that [[Subset|inclusion]], that is the [[binary relation]] of one set being a subset of another, is a [[partial order]].

'''PROPOSITION 6''': If {{tmath|1= A }}, {{tmath|1= B }} and {{tmath|1= C }} are sets then the following hold:
: [[reflexive relation|reflexivity]]:
:* {{tmath|1= A \subseteq A }}
: [[antisymmetric relation|antisymmetry]]:
:* {{tmath|1= A \subseteq B }} and {{tmath|1= B \subseteq A }} if and only if {{tmath|1= A = B }}
: [[transitive relation|transitivity]]:
:* If {{tmath|1= A \subseteq B }} and {{tmath|1= B \subseteq C }}, then {{tmath|1= A \subseteq C}}

The following proposition says that for any set ''S'', the [[power set]] of ''S'', ordered by inclusion, is a [[lattice (order)|bounded lattice]], and hence together with the distributive and complement laws above, show that it is a [[Boolean algebra (structure)|Boolean algebra]].

'''PROPOSITION 7''': If {{tmath|1= A }}, {{tmath|1= B }} and {{tmath|1= C }} are subsets of a set {{tmath|1= S }} then the following hold:
: existence of a [[greatest element|least element]] and a [[greatest element]]:
:* {{tmath|1= \varnothing \subseteq A \subseteq S }}
: existence of [[lattice (order)|joins]]:
:* {{tmath|1= A \subseteq A \cup B }}
:* If {{tmath|1= A \subseteq C }} and {{tmath|1= B \subseteq C }}, then {{tmath|1= A \cup B \subseteq C }}
: existence of [[lattice (order)|meets]]:
:* {{tmath|1= A \cap B \subseteq A }}
:* If {{tmath|1= C \subseteq A }} and {{tmath|1= C \subseteq B }}, then {{tmath|1= C \subseteq A \cap B }}

The following proposition says that the statement {{tmath|1= A \subseteq B }} is equivalent to various other statements involving unions, intersections and complements.

'''PROPOSITION 8''': For any two sets {{tmath|1= A }} and {{tmath|1= B }}, the following are equivalent:
:* {{tmath|1= A \subseteq B }}
:* {{tmath|1= A \cap B = A }}
:* {{tmath|1= A \cup B = B }}
:* {{tmath|1= A \setminus B = \varnothing }}
:* {{tmath|1= B^\complement \subseteq A^\complement }}

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous.