Given two sets A and B, A is a supersetMathworldPlanetmath of B if every element in B is also in A. We denote this relationMathworldPlanetmathPlanetmath as AB. This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that B is a subset of A, that is ABBA.

Similar rules to those that hold for also hold for . If XY and YX, then X=Y. Every set is a superset of itself, and every set is a superset of the empty setMathworldPlanetmath.

We say A is a proper superset of B if AB and AB. This relation is sometimes denoted by AB, but AB is often used to mean the more general superset relation, so it should be made explicit when “proper superset” is intended, possibly by using XY or XY (or XY or XY).

One will occasionally see a collectionMathworldPlanetmath C of subsets of some set X made into a partial orderMathworldPlanetmath “by containment”. Depending on context this can mean defining a partial order where YZ means YZ, or it can mean defining the opposite partial order: YZ means YZ. 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 categoryMathworldPlanetmath, usually by defining a single abstract morphismMathworldPlanetmath YZ whenever YZ (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