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Cardinal Addition


Let A and B be any sets with empty intersection, and let |X| denote the cardinal number of a set X. Then

 |A|+|B|=|A union B|

(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Rubin 1967, p. 274; Suppes 1972, pp. 112-113).

It is an interesting exercise to show that cardinal addition is well-defined. The main steps are to show that for any cardinal numbers a and b, there exist disjoint sets A and B with cardinal numbers a and b, and to show that if A and B are disjoint and C and D disjoint with |A|=|C| and |B|=|D| then |A union B|=|C union D|. The second of these is easy. The first is a little tricky and requires an appeal to the axioms of set theory. Also, one needs to restrict the definition of cardinal to guarantee if a is a cardinal, then there is a set A satisfying |A|=a.


See also

Cardinal Multiplication, Cardinal Exponentiation

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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.Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.

Referenced on Wolfram|Alpha

Cardinal Addition

Cite this as:

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

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