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# superset

Given two sets $A$ and $B$, $A$ is a *superset* of $B$ if every element in $B$ is also in $A$. We denote this relation^{} as $A\supseteq B$. This is equivalent^{} to saying that $B$ is a subset of $A$, 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 $X=Y$. Every set is a superset of itself, and every set is a superset of the empty set.

We say $A$ is a *proper superset* of $B$ 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 $C$ of subsets of some set $X$ 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 $C$ of subsets of some set $X$ 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.

## Mathematics Subject Classification

03E99*no label found*

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