Cardinality of Set of Injections/Formal Proof

Theorem

Let $S$ and $T$ be finite sets.

The number of injections from $S$ to $T$, where $\card S = m, \card T = n$ is often denoted ${}^m P_n$, and is:

${}^m P_n = \begin{cases} \dfrac {n!} {\paren {n - m}!} & : m \le n \\ 0 & : m > n \end{cases}$

Proof

Let $m > n$.

By the Pigeonhole Principle, there can be no injection from $S$ to $T$ when $\left|{S}\right| > \left|{T}\right|$.

Once $\left|{T}\right|$ 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 $\varnothing \to T$ is $\varnothing \times T$ which is $\varnothing$.

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 $k \in \left[{1 \,.\,.\, n}\right]$, let $\mathbb H \left({k, n}\right)$ be the set of all injections from $\N_k$ to $\N_n$.

The proof now proceeds by induction.

Let:

$\displaystyle \mathbb S = \left\{{k \in \left[{1 \,.\,.\, n}\right]: \left|{\mathbb H \left({k, n}\right)}\right| = \frac {n!} {\left({n - k}\right)!}}\right\}$

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:

$\displaystyle \left|{\mathbb H \left({1, n}\right)}\right| = n = \frac {n!} {\left({n - 1}\right)!}$

and so it follows that:

$1 \in \mathbb S$.

This is the basis for the induction.

Induction Hypothesis

We suppose that:

$\displaystyle \left|{\mathbb H \left({k, n}\right)}\right| = \frac {n!} {\left({n - k}\right)!}$.

This is the induction hypothesis.

We need to show that:

$\displaystyle \left|{\mathbb H \left({k+1, n}\right)}\right| = \frac {n!} {\left({n - \left({k+1}\right)}\right)!}$.

Induction Step

This is the induction step:

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

Let $\rho: \mathbb H \left({k + 1, n}\right) \to \mathbb H \left({k, n}\right)$ be the mapping defined by:

$\forall f \in \mathbb H \left({k + 1, n}\right): \rho \left({f}\right) =$ the restriction of $f$ to $\N_k$

Given that $g \in \mathbb H \left({k, n}\right)$ and $a \in \N_n - g \left({\N_k}\right)$, let $g_a: \N_{k + 1} \to \N_n$ be the mapping defined as:

$g_a \left({x}\right) = \begin{cases} g \left({x}\right) & : x \in \N_k \\ a & : x = k \end{cases}$

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

Hence:

$g_a \in \mathbb H \left({k + 1, n}\right)$

It follows from the definition of $\rho$ that:

$\rho^{-1} \left({\left\{{g}\right\}}\right) = \left\{{g_a: a \in \N_n - g \left({\N_k}\right)}\right\}$

Since $g$ is an injection, $g \left({\N_k}\right)$ has $k$ elements.

Therefore $\N_n - g \left({\N_k}\right)$ has $n - k$ elements by Cardinality of Complement.

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

Clearly:

$\left\{{\rho^{-1} \left({\left\{{g}\right\}}\right): g \in \mathbb H \left({k, n}\right)}\right\}$

is a partition of $\mathbb H \left({k + 1, n}\right)$.

Therefore by Number of Elements in Partition:

$\left|{\mathbb H \left({k + 1, n}\right)}\right| = \left({n - k}\right) \dfrac {n!} {\left({n - k}\right)!} = \dfrac {n!} {\left({\left({n - k}\right) - 1}\right)!}$

as $k \in \mathbb S$.

But:

$\left({n - k}\right) - 1 = n - \left({k + 1}\right)$

So:

$k + 1 \in \mathbb S$

By induction:

$\mathbb S = \left[{1 \,.\,.\, n}\right]$

and in particular:

$m \in \mathbb S$

$\blacksquare$