examples of ring of sets

Every field of sets is a ring of sets. Below are some examples of rings of sets that are not fields of sets.

  1. 1.

    Let A be a non-empty set containing an element a. Let be the family of subsets of A containing a. Then is a ring of sets, but not a field of sets, since {a}, but A-{a}.

  2. 2.

    The collectionMathworldPlanetmath of all open sets of a topological spaceMathworldPlanetmath is a ring of sets, which is in general not a field of sets, unless every open set is also closed. Likewise, the collection of all closed setsPlanetmathPlanetmath of a topological space is also a ring of sets.

  3. 3.

    A simple example of a ring of sets is the subset {{a},{a,b}} of 2{a,b}. That this is a ring of sets follows from the observations that {a}{a,b}={a} and {a}{a,b}={a,b}. Note that it is not a field of sets because the complement of {a}, which is {b}, does not belong to the ring.

  4. 4.

    Another example involves an infinite setMathworldPlanetmath. Let A be an infinite set. Let be the collection of finite subsets of A. Since the union and the intersectionMathworldPlanetmathPlanetmath of two finite setMathworldPlanetmath are finite sets, is a ring of sets. However, it is not a field of sets, because the complement of a finite subset of A is infiniteMathworldPlanetmath, and thus not a member of .

Title examples of ring of sets
Canonical name ExamplesOfRingOfSets
Date of creation 2013-03-22 15:47:52
Last modified on 2013-03-22 15:47:52
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Example
Classification msc 03E20
Classification msc 28A05