Given two sets A and B, we say that A is a subset of B (which we denote as AB or simply AB) if every element of A is also in B. That is, the following implicationMathworldPlanetmath holds:


The relationMathworldPlanetmath between A and B is then called set inclusion.

Some examples:

The set A={d,r,i,t,o} is a subset of the set B={p,e,d,r,i,t,o} because every element of A is also in B. That is, AB.

On the other hand, if C={p,e,d,r,o}, then neither AC (because tA but tC) nor CA (because pC but pA). The fact that A is not a subset of C is written as AC. Similarly, we have CA.

If XY and YX, it must be the case that X=Y.

Every set is a subset of itself, and the empty setMathworldPlanetmath is a subset of every other set. The set A is called a proper subsetMathworldPlanetmathPlanetmath of B, if AB and AB. In this case, we do not use AB.

Title subset
Canonical name Subset
Date of creation 2013-03-22 11:52:38
Last modified on 2013-03-22 11:52:38
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 13
Author Wkbj79 (1863)
Entry type Definition
Classification msc 03-00
Classification msc 00-02
Related topic EmptySet
Related topic SupersetMathworldPlanetmath
Related topic TotallyBounded
Related topic ProofThatAllSubgroupsOfACyclicGroupAreCyclic
Related topic Property2
Related topic CardinalityOfAFiniteSetIsUnique
Related topic CriterionOfSurjectivity
Defines set inclusion