%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: superset
%%% Primary Category Code: 03E99
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%%% Version: 9
%%% Owner: yark
%%% Author(s): yark, archibal, Logan
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Given two sets $A$ and $B$, $A$ is a \emph{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 \htmladdnormallink{subset}{http://planetmath.org/encyclopedia/ProperSubset2.html} 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 \htmladdnormallink{empty set}{http://planetmath.org/encyclopedia/EmptySet.html}.

We say $A$ is a \emph{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 \htmladdnormallink{collection}{http://planetmath.org/encyclopedia/SetOfSets.html} $C$ of subsets of some set $X$ made into a \htmladdnormallink{partial order}{http://planetmath.org/encyclopedia/PartialOrdering.html} ``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 \htmladdnormallink{opposite}{http://planetmath.org/encyclopedia/OppositeSide.html} partial order: $Y\leq Z$ means $Y \supseteq Z$. This is frequently used when applying \htmladdnormallink{Zorn's lemma}{http://planetmath.org/encyclopedia/ZornsLemma.html}.

One will also occasionally see a collection $C$ of subsets of some set $X$ made into a \htmladdnormallink{category}{http://planetmath.org/encyclopedia/Identity2.html}, usually by defining a single abstract \htmladdnormallink{morphism}{http://planetmath.org/encyclopedia/StructureAutomorphism.html} $Y\to Z$ whenever $Y\subseteq Z$ (this being a special case of the general method of treating \htmladdnormallink{pre-orders}{http://planetmath.org/encyclopedia/Quasiordered2.html} as categories). This allows a concise definition of \htmladdnormallink{presheaves}{http://planetmath.org/encyclopedia/Sheaf2.html} and \htmladdnormallink{sheaves}{http://planetmath.org/encyclopedia/Sheaf2.html}, and it is generalized when defining a \htmladdnormallink{site}{http://planetmath.org/encyclopedia/Covering.html}.

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