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## Proposition: Number of Ordered n-Tuples in a Set

Let $X$ be a non-empty set and let $\{a_1,a_2,\ldots,a_n\}$ be a set with $n$ elements. The Cartesian product $$\underbrace{X\times X\times\cdots\times X}_{n\text{ times}}$$ is equipotent to the set of all maps $f:\{a_1,a_2,\ldots,a_n\}\to X.$

In particular, if $X$ is a finite set as well with the cardinality $|X|=m,$ then the number of ordered n-tuples of the $m$ elements equals $m^n.$

| | | | | created: 2019-09-07 08:37:40 | modified: 2019-09-07 08:37:40 | by: bookofproofs

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