subset
Given two sets and , we say that is a subset of (which we denote as or simply ) if every element of is also in . That is, the following implication holds:
The relation between and is then called set inclusion.
Some examples:
The set is a subset of the set because every element of is also in . That is, .
On the other hand, if , then neither (because but ) nor (because but ). The fact that is not a subset of is written as . Similarly, we have .
If and , it must be the case that .
Every set is a subset of itself, and the empty set is a subset of every other set. The set is called a proper subset of , if and . In this case, we do not use .
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 | Superset |
Related topic | TotallyBounded |
Related topic | ProofThatAllSubgroupsOfACyclicGroupAreCyclic |
Related topic | Property2 |
Related topic | CardinalityOfAFiniteSetIsUnique |
Related topic | CriterionOfSurjectivity |
Defines | set inclusion |