# Cartesian Product is Anticommutative

## Theorem

Let $S, T \ne \O$.

Then:

$S \times T = T \times S \implies S = T$

### Corollary

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

## Proof

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

Then:

 $\displaystyle x \in S$ $\land$ $\displaystyle y \in T$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \tuple {x, y}$ $\in$ $\displaystyle S \times T$ Definition of Cartesian Product $\displaystyle \leadstoandfrom \ \$ $\displaystyle \tuple {x, y}$ $\in$ $\displaystyle T \times S$ by hypothesis $\displaystyle \leadstoandfrom \ \$ $\displaystyle x \in T$ $\land$ $\displaystyle y \in S$ Definition of Cartesian Product

Thus it can be seen from the definition of set equality that $S \times T = T \times S \implies S = T$.

Note that if $S = \O$ or $T = \O$ then, from Cartesian Product is Empty iff Factor is Empty, $S \times T = T \times S = \O$ whatever $S$ and $T$ are, and the result does not hold.

$\blacksquare$

## Sources

except that the case where either set is empty has been ignored