Cardinality of Set of Injections/Formal Proof

Proof
Let $m > n$.

By the Pigeonhole Principle, there can be no injection from $S$ to $T$ when $\card S > \card T$.

Once $\card T$ elements of $S$ have been used up, there is no element of $T$ left for the remaining elements of $S$ to be mapped to such that they all still map to different elements of $T$.

Let $m = 0$.

The only injection from $\O \to T$ is $\O \times T$ which is $\O$.

So if $m = 0$ there is $1 = n! / n!$ injection.

Let $0 < m \le n$.

As in the proof of Cardinality of Set of All Mappings, we can assume that $S = \N_m$ and $T = \N_n$.

For each $k \in \closedint 1 n$, let $\map {\mathbb H} {k, n}$ be the set of all injections from $\N_k$ to $\N_n$.

The proof now proceeds by induction.

Let:


 * $\ds \mathbb S = \set {k \in \closedint 1 n: \card {\map {\mathbb H} {k, n} } = \frac {n!} {\paren {n - k}!} }$

Basis for the Induction
Let $k = 1$.

From Cardinality of Set of All Mappings, there are $n^1 = n$ different mappings from $S$ to $T$.

From Mapping from Singleton is Injection, each one of these $n$ mappings is an injection.

Thus:
 * $\ds \card {\map {\mathbb H} {1, n} } = n = \frac {n!} {\paren {n - 1}!}$

and so it follows that:
 * $1 \in \mathbb S$.

This is the basis for the induction.

Induction Hypothesis
We suppose that:
 * $\ds \card {\map {\mathbb H} {k, n} } = \frac {n!} {\paren {n - k}!}$.

This is the induction hypothesis.

We need to show that:
 * $\ds \card {\map {\mathbb H} {k + 1, n} } = \frac {n!} {\paren {n - \paren {k + 1} }!}$.

Induction Step
This is the induction step:

Let $k \in \mathbb S$ such that $k < n$.

Let $\rho: \map {\mathbb H} {k + 1, n} \to \map {\mathbb H} {k, n}$ be the mapping defined by:
 * $\forall f \in \map {\mathbb H} {k + 1, n}: \map \rho f =$ the restriction of $f$ to $\N_k$

Given that $g \in \map {\mathbb H} {k, n}$ and $a \in \N_n - \map g {\N_k}$, let $g_a: \N_{k + 1} \to \N_n$ be the mapping defined as:
 * $\map {g_a} x = \begin {cases} \map g x & : x \in \N_k \\ a & : x = k \end {cases}$

Now $g$ is an injection as $g \in \map {\mathbb H} {k, n}$, and as $\map {g_a} a \notin \map g {\N_k}$ it follows that $g_a$ is also an injection.

Hence:
 * $g_a \in \map {\mathbb H} {k + 1, n}$

It follows from the definition of $\rho$ that:
 * $\map {\rho^{-1} } {\set g} = \set {g_a: a \in \N_n - \map g {\N_k} }$

Since $g$ is an injection, $\map g {\N_k}$ has $k$ elements.

Therefore $\N_n - \map g {\N_k}$ has $n - k$ elements by Cardinality of Complement.

As $G: a \to g_a$ is clearly a bijection from $\N_n - \map g {\N_k}$ onto $\map {\rho^{-1} } {\set g}$, that set has $n - k$ elements.

Clearly:
 * $\set {\map {\rho^{-1} } {\set g}: g \in \map {\mathbb H} {k, n} }$

is a partition of $\map {\mathbb H} {k + 1, n}$.

Therefore by Number of Elements in Partition:
 * $\card {\map {\mathbb H} {k + 1, n} } = \paren {n - k} \dfrac {n!} {\paren {n - k}!} = \dfrac {n!} {\paren {\paren {n - k} - 1}!}$

as $k \in \mathbb S$.

But:
 * $\paren {n - k} - 1 = n - \paren {k + 1}$

So:
 * $k + 1 \in \mathbb S$

By induction:
 * $\mathbb S = \closedint 1 n$

and in particular:
 * $m \in \mathbb S$