# Product of Finite Sets is Finite/Proof 1

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## Theorem

Let $S$ and $T$ be finite sets.

Then $S \times T$ is a finite set.

## Proof

By the definition of Cartesian product:

- $S \times T = \left\{{ (s,t) } \middle\vert {s \in S, t \in T}\right\}$

Then by definition of set union:

- $S \times T = \displaystyle \bigcup_{s \in S} \{s\} \times T$

Also, the mapping $g_s: \{s\} \times T \to T$ defined by:

- $g_s \left({s, t}\right) = t$

is a bijection.

Therefore, since $T$ is finite, so is $\{s\} \times T$ for all $s \in S$.

Since $S$ is finite, the result follows from Finite Union of Finite Sets is Finite.

$\blacksquare$

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 6$: Finite Sets: Corollary $6.8$