Cardinality of Set of All Mappings

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Theorem

Let $S$ and $T$ be sets.

The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is:

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


Proof for Finite Sets

Let $\card S = n$ and $\card T = m$.


First suppose that $n = 0$, that is, that $S = \O$.

From Cardinality of Set of All Mappings from Empty Set:

$\card {T^\O} = 1 = m^0$

and the result is seen to hold for $n = 0$.


Next, suppose that $m = 0$, that is, that $T = \O$.

From Cardinality of Set of All Mappings to Empty Set:

$\card {\O^S} = \begin{cases} 1 & : S = \O \\ 0 & : S \ne \O \end{cases}$

So if $n > 0$:

$\card {\O^S} = 0 = 0^n$

and if $n = 0$:

$\card {T^S} = 1 = 0^0 = m^n$

and the result holds.


This fits in with the preferred definition of the value of $0^0$.


Finally, suppose $m > 0$ and $n > 0$.

Let $\sigma: \N_n \to S$ and $\tau: T \to \N_n$ be bijections.

Then the mapping $\Phi: T^S \to \paren {\N_m}^{\paren {\N_n} }$ defined as:

$\forall f \in T^S: \Phi \paren f = \tau \circ f \circ \sigma$

(where $\paren {\N_m}^{\paren {\N_n} }$ is the set of all mappings from $\N_n$ to $\N_m$) is also a bijection.

So we need only consider the case where $S = \N_n$ and $T = \N_m$.


Let $m \in \N_{>0}$.

For each $n \in \N$, let $\Bbb T \paren {n, m}$ be the set of all mappings from $\N_n$ to $\N_m$.

Let:

$\Bbb S = \set {n \in \N: \card {\Bbb T \tuple {n, m} } = m^n}$

We have seen that $0 \in \Bbb S$.

Let $n \in \Bbb S$.

Let $\rho: \Bbb T \tuple {n + 1, m} \to \Bbb T \paren {n, m}$ defined by:

$\forall f \in \Bbb T \tuple {n + 1, m}: \rho \paren f =$ the restriction of $f$ to $\N_n$

Given that $g \in \Bbb T \tuple {n, m}$, and $k \in \N_m$, let $g_k: \N_{n + 1} \to \N_m$ be defined by:

$\forall x \in \N_{n + 1}: g_k \paren x = \begin{cases} g \paren x & : x \in \N_n \\ k & : x = n \end{cases}$

Then:

$\rho^{-1} \paren {\set g} = \set {g_0, \ldots, g_{m-1} }$

Thus $\rho^{-1} \paren {\set g}$ has $m$ elements.

So clearly:

$\set {\rho^{-1} \paren {\set g}: g \in \Bbb T \tuple {n, m} }$

is a partition of $\Bbb T \tuple {n + 1, m}$.

Hence, as $n \in \Bbb S$, the set $\Bbb T \tuple {n + 1, m}$ has $m \cdot m^n = m^{n + 1}$ elements by Number of Elements in Partition.

Thus $n + 1 \in \Bbb S$.

By the Principle of Mathematical Induction:

$\Bbb S = \N$

and the proof is complete.

$\blacksquare$


Proof for Infinite Sets


Examples

$3$ Elements to $2$ Elements

Let $S = \set {1, 2, 3}$.

Let $T = \set {a, b}$.

Then the mappings from $S$ to $T$ in two-row notation are:

$\dbinom {1 \ 2 \ 3} {a \ a \ a}, \dbinom {1 \ 2 \ 3} {a \ a \ b}, \dbinom {1 \ 2 \ 3} {a \ b \ a}, \dbinom {1 \ 2 \ 3} {a \ b \ b}, \dbinom {1 \ 2 \ 3} {b \ a \ a}, \dbinom {1 \ 2 \ 3} {b \ a \ b}, \dbinom {1 \ 2 \ 3} {b \ b \ a}, \dbinom {1 \ 2 \ 3} {b \ b \ b}$

a total of $2^3 = 8$.


All but the first and last are surjections.

None are injections.


Comment

The question of whether to define $0^0 = 0$ or $0^0 = 1$ keeps students awake arguing for hours.

Here's another argument, in case you're not convinced, for defining $0^0 = 1$ as opposed to $0^0 = 0$ - another result kept nice and neat.