# Cartesian Product is Anticommutative

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## Contents

## 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.

$\blacksquare$

## Also see

## Sources

- 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 8$. Notations and definitions of set theory

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

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.2$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets: Exercise $\text{C} \ 1$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $1$. Sets: Exercise $8$

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

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 9$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.9$: Cartesian Product - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 5$: Products of Sets - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $13$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Exercise $11$