If , are cardinals, define to be the cardinal number of the set of
Question:
If α, β are cardinals, define αβ to be the cardinal number of the set of all functions B → A, where A,B are sets such that IAI = α, IBI = β
(a) αβ is independent of the choice of A,B.
(b) αβ+ϒ = (αβ}(αϒ); (αβ)ϒ = (αϒ)(βϒ); αβϒ = (αβ)ϒ
(c) If α ≤ β, then αϒ ≤βϒ.
(d) If α,β are finite with α > 1, β > l and ϒ is infinite, then αϒ = βϒ.
(e) For every finite cardinal n, αn = α.α··· α (n factors). Hence αn = α if α. is infinite.
(f) If P(A) is the power set of a set A, then IP(A)I = 2IAI.
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a To show that is independent of the choice of AB we need to show that there exists a bijection between the set of functions from B to A with IAI IBI ...View the full answer
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Related Book For
Algebra Graduate Texts In Mathematics 73
ISBN: 9780387905181
8th Edition
Authors: Thomas W. Hungerford
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