Similar rules to those that hold for also hold for . If and , then . Every set is a superset of itself, and every set is a superset of the empty set.
We say is a proper superset of if and . This relation is sometimes denoted by , but is often used to mean the more general superset relation, so it should be made explicit when “proper superset” is intended, possibly by using or (or or ).
One will occasionally see a collection of subsets of some set made into a partial order “by containment”. Depending on context this can mean defining a partial order where means , or it can mean defining the opposite partial order: means . This is frequently used when applying Zorn’s lemma.
One will also occasionally see a collection of subsets of some set made into a category, usually by defining a single abstract morphism whenever (this being a special case of the general method of treating pre-orders as categories). This allows a concise definition of presheaves and sheaves, and it is generalized when defining a site.
|Date of creation||2013-05-24 14:35:12|
|Last modified on||2013-05-24 14:35:12|
|Last modified by||unlord (1)|