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Viewing Version
5
of
'superset'
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| Title of object: |
superset |
| Canonical Name: |
Superset |
| Type: |
Definition |
| Created on: |
2002-02-24 04:26:13 |
| Modified on: |
2006-09-09 02:33:12 |
| Classification: |
msc:03E99 |
| Defines: |
proper superset, contains, contained |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts} |
Content:
\PMlinkescapeword{equivalent}
\PMlinkescapeword{relation}
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 often denoted by $A \supset B$. Unfortunately, $A \supset B$ is often used to \PMlinkescapetext{mean} the more general superset relation, and thus it should be made explicit when ``proper superset'' is intended, possibly by using $X\supsetneq Y$.
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 allows a concise definition of presheaves and sheaves, and it is generalized when defining a site. |
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