# Cardinality of Cartesian Product/Corollary

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## Corollary to Cardinality of Cartesian Product

Let $S \times T$ be the cartesian product of two sets $S$ and $T$ which are both finite.

Then:

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

where $\card {S \times T}$ denotes the cardinality of $S \times T$.

## Proof 1

\(\displaystyle \left\vert{S \times T}\right\vert\) | \(=\) | \(\displaystyle \left\vert{S}\right\vert \times \left\vert{T}\right\vert\) | Cardinality of Cartesian Product | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left\vert{T}\right\vert \times \left\vert{S}\right\vert\) | Integer Multiplication is Commutative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left\vert{T \times S}\right\vert\) | Cardinality of Cartesian Product |

$\blacksquare$

## Proof 2

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

- $\forall \left({s, t}\right) \in S \times T: f \left({s, t}\right) = \left({t, s}\right)$

which is shown to be bijective as follows:

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

\(\displaystyle \implies \ \ \) | \(\displaystyle \left({t_1, s_1}\right)\) | \(=\) | \(\displaystyle \left({t_2, s_2}\right)\) | Definition of $f$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \left({s_1, t_1}\right)\) | \(=\) | \(\displaystyle \left({s_2, t_2}\right)\) | Equality of Ordered Pairs |

showing $f$ is an injection.

Let $\left({t, s}\right) \in T \times S$.

Then:

- $\exists \left({s, t}\right) \in S \times T: f \left({s, t}\right) = \left({t, s}\right)$

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

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $3$. Mappings: Exercise $9 \ \text {(i)}$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products: Exercise $1$