Product of Finite Sets is Finite

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Theorem

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


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


Proof 1

By the definition of Cartesian product:

$S \times T = \set {\tuple {s, t}: s \in S, t \in T}$

Then by definition of set union:

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

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

$\map {g_s} {s, t} = t$

is a bijection.

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

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

$\blacksquare$


Proof 2

Let $\card S$ denote the cardinal number of $S$.

Let $\cdot$ denote ordinal multiplication.


By Cardinal Product Equinumerous to Ordinal Product, it follows that $S \times T \sim \card S \cdot \card T$.

But then $\card S$ and $\card T$ are members of the minimal infinite successor set.

Therefore, $\card S \cdot \card T \in \omega$ by Natural Number Multiplication is Closed.


Since $S \times T$ is equinumerous to a member of the minimal infinite successor set, it follows that $S \times T$ is finite.

$\blacksquare$