Cardinal Product Equal to Maximum

Theorem

Let $S$ and $T$ be sets that are equinumerous to their cardinal number.

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

Suppose $S$ is infinite.

Suppose $T > 0$.

Then:

$\card {S \times T} = \map \max {\card S, \card T}$

Proof

Let $x$ denote $\map \max {\card S, \card T}$.

Then by Cartesian Product Preserves Cardinality:

$S \times T \sim \card S \times \card T$

Let $f: S \times T \to \card S \times \card T$ be a bijection.

It follows that $f: S \times T \to x \times x$ is an injection.

Hence:

\(\ds \card {S \times T}\)

\(\le\)

\(\ds \card {x \times x}\)

Injection iff Cardinal Inequality

\(\ds \)

\(=\)

\(\ds \card x\)

Non-Finite Cardinal is equal to Cardinal Product

\(\ds \)

\(\le\)

\(\ds x\)

Cardinal Number Less than Ordinal: Corollary

Therefore:

$\card {S \times T} \le x$

$\Box$

Conversely:

$x = \card S$ if $\card T \le \card S$

and:

$x = \card T$ if $\card S \le \card T$

By Relation between Two Ordinals:

$x = \card S$ or $x = \card T$

It follows by Set Less than Cardinal Product that:

$x \le \card {S \times T}$

$\Box$

Combining the two lemmas, it follows that:

$x = \card {S \times T}$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.35$