Editing Subset (section)

== Other properties of inclusion ==
* A  set ''A'' is a '''subset''' of ''B'' [[if and only if]] their intersection is equal to A. Formally:
:<math> A \subseteq B \text{ if and only if } A \cap B = A. </math>
* A  set ''A'' is a '''subset''' of ''B'' if and only if their union is equal to B. Formally:
:<math> A \subseteq B \text{ if and only if } A \cup B = B. </math>
* A '''finite''' set ''A'' is a '''subset''' of ''B'', if and only if the [[cardinality]] of their intersection is equal to the cardinality of A. Formally:
:<math> A \subseteq B \text{ if and only if } |A \cap B| = |A|.</math>
* The subset relation defines a [[partial order]] on sets. In fact, the subsets of a given set form a [[Boolean algebra (structure)|Boolean algebra]] under the subset relation, in which the [[join and meet]] are given by [[Intersection (set theory)|intersection]] and [[Union (set theory)|union]], and the subset relation itself is the [[Inclusion (Boolean algebra)|Boolean inclusion relation]].
* Inclusion is the canonical [[partial order]], in the sense that every partially ordered set <math>(X, \preceq)</math> is [[isomorphic]] to some collection of sets ordered by inclusion. The [[ordinal number]]s are a simple example: if each ordinal ''n'' is identified with the set <math>[n]</math> of all ordinals less than or equal to ''n'', then <math>a \leq b</math> if and only if <math>[a] \subseteq [b].</math>