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 $A\supseteq B\iff B\subseteq A$.

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 $A\ne B$. 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 $X⊋Y$ or $X⫌Y$ (or $X⊋Y$ or $X⫌Y$).

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 $Y\le Z$ means $Y\subseteq Z$, or it can mean defining the opposite partial order: $Y\le Z$ 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.

Title	superset
Canonical name	Superset
Date of creation	2013-05-24 14:35:12
Last modified on	2013-05-24 14:35:12
Owner	yark (2760)
Last modified by	unlord (1)
Numerical id	13
Author	yark (1)
Entry type	Definition
Classification	msc 03E99
Related topic	Subset
Related topic	SetTheory
Defines	proper superset
Defines	contains
Defines	contained