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单词 CardinalArithmetic
释义

cardinal arithmetic


Definitions

Let κ and λ be cardinal numbersMathworldPlanetmath,and let A and B be disjoint sets such that |A|=κ and |B|=λ.(Here |X| denotes the cardinality of a set X,that is, the unique cardinal number equinumerous with X.)Then we define cardinal additionMathworldPlanetmath, cardinal multiplicationand cardinal exponentiation as follows.

κ+λ=|AB|.
κλ=|A×B|.
κλ=|AB|.

(Here AB denotes the set of all functions from B to A.)These three operationsMathworldPlanetmath are well-defined, that is,they do not depend on the choice of A and B.Also note that for multiplication and exponentiation Aand B do not actually need to be disjoint.

We also define addition and multiplication for arbitrary numbers of cardinals.Suppose I is an index setMathworldPlanetmathPlanetmath and κi is a cardinal for every iI.Then iIκi is defined to bethe cardinality of the union iIAi,where the Ai are pairwise disjoint and |Ai|=κi for each iI.Similarly, iIκi is defined to be the cardinality of theCartesian product (http://planetmath.org/GeneralizedCartesianProduct)iIBi, where |Bi|=κi for each iI.

Properties

In the following, κ, λ, μ and ν are arbitrary cardinals,unless otherwise specified.

Cardinal arithmetic obeys many of the same algebraic laws as real arithmeticPlanetmathPlanetmath.In particular, the following properties hold.

κ+λ=λ+κ.
(κ+λ)+μ=κ+(λ+μ).
κλ=λκ.
(κλ)μ=κ(λμ).
κ(λ+μ)=κλ+κμ.
κλκμ=κλ+μ.
(κλ)μ=κλμ.
κμλμ=(κλ)μ.

Some special cases involving 0 and 1 are as follows:

κ+0=κ.
0κ=0.
κ0=1.
0κ=0, for κ>0.
1κ=κ.
κ1=κ.
1κ=1.

If at least one of κ and λ is infiniteMathworldPlanetmath, then the following hold.

κ+λ=max(κ,λ).
κλ=max(κ,λ), provided κ0λ.

Also notable is that if κ and λ are cardinalswith λ infinite and 2κ2λ,then

κλ=2λ.

Inequalities are also important in cardinal arithmetic.The most famous is Cantor’s theoremMathworldPlanetmath

κ<2κ.

If μκ and νλ, then

μ+νκ+λ.
μνκλ.
μνκλ, unless μ=ν=κ=0<λ.

Similar inequalities hold for infinite sums and productsMathworldPlanetmath.Let I be an index set,and suppose that κi and λi are cardinals for every iI.If κiλi for every iI, then

iIκiiIλi.
iIκiiIλi.

If, moreover, κi<λi for all iI,then we have König’s theorem.

iIκi<iIλi.

If κi=κ for every i in the index set I, then

iIκi=κ|I|.
iIκi=κ|I|.

Thus it is possible to define exponentiation in terms of multiplication,and multiplication in terms of addition.

Titlecardinal arithmetic
Canonical nameCardinalArithmetic
Date of creation2013-03-22 14:15:13
Last modified on2013-03-22 14:15:13
Owneryark (2760)
Last modified byyark (2760)
Numerical id38
Authoryark (2760)
Entry typeTopic
Classificationmsc 03E10
Related topicOrdinalArithmetic
Related topicCardinalNumber
Related topicCardinalExponentiationUnderGCH
Related topicCardinalityOfTheContinuum
Definescardinal addition
Definescardinal multiplication
Definescardinal exponentiation
Definessum of cardinals
Definesproduct of cardinals
Definesaddition
Definesmultiplication
Definesexponentiation
Definessum
Definesproduct
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