properties of cardinal numbers


Theorem. Let (cγ)γΓ be an indexed family of cardinal numbersMathworldPlanetmath indexed by a nonempty index setMathworldPlanetmath Γ. Also, let (Γδ)δΔ be an arbitrary indexed partitionMathworldPlanetmathPlanetmath of the index set. Then we have the following properties:

1. Associative Laws.

γΓcγ=δΔγΓδcγ

and

γΓcγ=δΔγΓδcγ.

2. Commutative Laws. Let π:ΓΓ be a partition. Then

γΓcγ=γΓcπ(γ)

and

γΓcγ=γΓcπ(γ).

3. Distributive Laws. Let a be any arbitrary infiniteMathworldPlanetmath cardinal number. Then

a(γΓcγ)=γΓacγ
Title properties of cardinal numbers
Canonical name PropertiesOfCardinalNumbers
Date of creation 2013-03-22 16:08:29
Last modified on 2013-03-22 16:08:29
Owner gilbert_51126 (14238)
Last modified by gilbert_51126 (14238)
Numerical id 7
Author gilbert_51126 (14238)
Entry type Theorem
Classification msc 03-00