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Cardinal Exponentiation

Let A and B be any sets, and let |X| be the cardinal number of a set X. Then cardinal exponentiation is defined by

|A|^(|B|)=|set of all functions from B into A|

(Ciesielski 1997, p. 68; Dauben 1990, p. 174; Moore 1982, p. 37; Rubin 1967, p. 275, Suppes 1972, p. 116).

It is easy to show that the cardinal number of the power set of A is 2^(|A|), since |{0,1}|=2 and there is a natural bijection between the subsets of A and the functions from A into {0,1}.

See also

Cardinal Addition, Cardinal Multiplication, Cardinal Number, Power Set

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References

Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

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Cardinal Exponentiation

Cite this as:

Weisstein, Eric W. "Cardinal Exponentiation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CardinalExponentiation.html

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