Cardinality of Set of Injections/Formal Proof

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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 $\card S > \card T$.

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:

$m \in \mathbb S$

$\blacksquare$


Sources

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