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Partial Order Collection Opposite Category Set Empty Set Sheaf Quasi Order Site Structure Preserving Mappings Zorns Lemma

1	Category
1	Partial Order
1	Opposite
1	Collection
1	Empty Set
1	Set
2	Zorns Lemma
2	Quasi Order
2	Sheaf
2	Site
2	Structure Preserving Mappings

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Superset

Given two sets and , is a superset of if every element in is also in . We denote this relation as $A\supseteq B$ . This is equivalent to saying that is a subset of , that is $A\supseteq B \Leftrightarrow B\subseteq A$ .

Similar rules to those that hold for $\subseteq$ also hold for $\supseteq$ . If $X\supseteq Y$ and $Y\supseteq X$ , 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 $A \supseteq B$ and $A \neq B$ . This relation is sometimes denoted by $A \supset B$ , but $A \supset B$ is often used to mean the more general superset relation, so it should be made explicit when “proper superset” is intended, possibly by using $X\varsupsetneq Y$ or $X\supsetneqq Y$ (or $X\supsetneq Y$ or $X\varsupsetneqq Y$ ).

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 $Y\leq Z$ means $Y \subseteq Z$ , or it can mean defining the opposite partial order: $Y\leq Z$ means $Y \supseteq Z$ . 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 $Y\to Z$ whenever $Y\subseteq Z$ (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.