# Cartesian Product is Anticommutative/Corollary

## Corollary to Cartesian Product is Anticommutative

Let $S$ and $T$ be sets.

Then:

$S \times T = T \times S \iff S = T \lor S = \O \lor T = \O$

where $S \times T$ denotes the cartesian product of $S$ and $T$.

## Proof

Suppose $S \times T = T \times S$.

Then either:

$(1): \quad S \ne \O \land T \ne \O$ and from Cartesian Product is Anticommutative, $S = T$

or:

$(2): \quad S = \O \lor T = \O$ and from Cartesian Product is Empty iff Factor is Empty, $S \times T = T \times S = \O$.

In either case, we see that:

$S \times T = T \times S \implies S = T \lor S = \O \lor T = \O$

Now suppose $S = T \lor S = \O \lor T = \O$.

From Cartesian Product is Empty iff Factor is Empty, we have that:

$S = \O \lor T = \O \implies S \times T = \O = T \times S$

Similarly:

$S = T \land \neg \paren {S = \O \lor T = \O} \implies S \times T = T \times S$

by definition of equality.

$\blacksquare$