Algebra: Difference between revisions
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Let <math>X</math> be a nonempty set. An '''algebra''' <math>\mathcal{A} \subseteq 2^X</math> is a nonempty collection of subsets of <math>X</math> that is closed under finite unions and complements. | Let <math>X</math> be a nonempty set. An '''algebra'''<ref name="Folland">Folland, Gerald B. (1999). ''Real Analysis: Modern Techniques and Their Applications'', John Wiley and Sons, ISBN 0471317160, Second edition.</ref> <math>\mathcal{A} \subseteq 2^X</math> is a nonempty collection of subsets of <math>X</math> that is closed under finite unions and complements. | ||
By DeMorgan's laws, an algebra is also closed under finite intersections | By DeMorgan's laws, an algebra is also closed under finite intersections and contains the empty set and <math>X</math>. | ||
==Examples of | ==Examples of Algebras== | ||
Assume that <math>X</math> is nonempty. | Assume that <math>X</math> is nonempty. | ||
* Given a set <math>X</math>, then <math>2^X</math> and <math>\{\emptyset,X\}</math> are algebras. | * Given a set <math>X</math>, then <math>2^X</math> and <math>\{\emptyset,X\}</math> are algebras. | ||
* Given a set <math>X</math>, the collection of all finite and cofinite (having finite complement) subsets of <math>X</math> is an algebra. | * Given a set <math>X</math>, the collection of all finite and cofinite (having finite complement) subsets of <math>X</math> is an algebra. | ||
* A [[<math>\sigma</math>-algebra]] is a | * A [[Sigma-algebra | <math>\sigma</math>-algebra]] is a more restrictive type of algebra. To show an algebra is a <math>\sigma</math>-algebra, it suffices to show closure under countable disjoint unions, which is notably not guaranteed by the definition of an algebra. | ||
==Non-examples== | ==Non-examples== | ||
* Given a topological space <math>(X,\tau)</math>, the topology <math>\tau</math> is in general not an algebra | * Given a topological space <math>(X,\tau)</math>, the topology <math>\tau</math> is in general not an algebra, for the complement of an open set <math>U \in \tau</math> may fail to be open. For example, in <math>X = \mathbb{R}</math> with the standard topology, the open interval <math>(0,1)</math> is open, but its complement <math>(0,1)^c = (-\infty,0] \cup [1,+\infty)</math> is not. | ||
==References== | |||
Latest revision as of 23:10, 13 November 2020
Let be a nonempty set. An algebra[1] is a nonempty collection of subsets of that is closed under finite unions and complements.
By DeMorgan's laws, an algebra is also closed under finite intersections and contains the empty set and .
Examples of Algebras
Assume that is nonempty.
- Given a set , then and are algebras.
- Given a set , the collection of all finite and cofinite (having finite complement) subsets of is an algebra.
- A -algebra is a more restrictive type of algebra. To show an algebra is a -algebra, it suffices to show closure under countable disjoint unions, which is notably not guaranteed by the definition of an algebra.
Non-examples
- Given a topological space , the topology is in general not an algebra, for the complement of an open set may fail to be open. For example, in with the standard topology, the open interval is open, but its complement is not.
![{\displaystyle (0,1)^{c}=(-\infty ,0]\cup [1,+\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3ba536257798c09f799bd29629be5a3509de43)
References
- ↑ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.