# 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 $X\u228bY$ or $X\u2accY$ (or $X\u228bY$ or $X\u2accY$).

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 |