Cartesian Product is Small iff Inverse is Small

Theorem

Let $A$ and $B$ be classes.

Then the Cartesian product $A \times B$ is a small class if and only if $B \times A$ is small.

Proof

\(\ds A \times B\)

\(=\)

\(\ds \set {\tuple {x, y} : x \in A \land y \in B}\)

Definition of Cartesian Product

\(\ds \)

\(=\)

\(\ds \set {\tuple {y, x} : x \in A \land y \in B}^{-1}\)

Definition of Inverse Relation

\(\ds \)

\(=\)

\(\ds \paren {B \times A}^{-1}\)

Definition of Cartesian Product

Let $B \times A$ be a small class.

Then, by Inverse of Small Relation is Small, $A \times B$ is also small.

Similarly, let $A \times B$ be small.

Then, by Inverse of Small Relation is Small, $B \times A$ is also small.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.9 \ (1)$