Cardinality of Cartesian Product of Finite Sets/Corollary/Proof 2

Corollary to Cardinality of Cartesian Product of Finite Sets

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

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:

$\ds \map f {s_1, t_1}$

$=$

$\ds \map f {s_2, t_2}$

$\ds \leadsto \ \ $

$\ds \tuple {t_1, s_1}$

$=$

$\ds \tuple {t_2, s_2}$

Definition of $f$

$\ds \leadsto \ \ $

$\ds \tuple {s_1, t_1}$

$=$

$\ds \tuple {s_2, t_2}$

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.

$\blacksquare$