# Cardinality of Cartesian Product/Corollary/Proof 2

Jump to navigation
Jump to search

## Corollary to Cardinality of Cartesian Product

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

## Proof

Let $f: S \times T \to T \times S$ be the mapping defined as:

- $\forall \tuple {s, t} \in S \times T: \map f {s, t} = \tuple {t, s}$

which is shown to be bijective as follows:

\(\displaystyle \map f {s_1, t_1}\) | \(=\) | \(\displaystyle \map f {s_2, t_2}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \tuple {t_1, s_1}\) | \(=\) | \(\displaystyle \tuple {t_2, s_2}\) | Definition of $f$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \tuple {s_1, t_1}\) | \(=\) | \(\displaystyle \tuple {s_2, t_2}\) | Equality of Ordered Pairs |

showing $f$ is an injection.

Let $\tuple {t, s} \in T \times S$.

Then:

- $\exists \tuple {s, t} \in S \times T: \map f {s, t} = \tuple {t, s}$

showing that $f$ is a surjection.

So we have demonstrated that there exists a bijection from $S \times T$ to $T \times S$.

The result follows by definition of set equivalence.

$\blacksquare$

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{P}$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 8$: Example $8.1$