superset Superset 2002-02-24 04:26:13 2006-09-17 09:34:48 Definition proper superset contains contained \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{equivalent} \PMlinkescapeword{mean} \PMlinkescapeword{relation} \PMlinkescapeword{similar} 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 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 \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 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.