Cardinality of Set of All Mappings/Examples

Examples of Cardinality of Set of All Mappings

$2$ Elements to $2$ Elements

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

Let $Y = \set {u, v}$.

Then the mappings from $X$ to $Y$ are:

\(\text {(1)}: \quad\)

\(\ds \map {f_1} a\)

\(=\)

\(\ds u\)

\(\ds \map {f_1} b\)

\(=\)

\(\ds v\)

\(\text {(2)}: \quad\)

\(\ds \map {f_2} a\)

\(=\)

\(\ds u\)

\(\ds \map {f_2} b\)

\(=\)

\(\ds u\)

\(\text {(3)}: \quad\)

\(\ds \map {f_3} a\)

\(=\)

\(\ds v\)

\(\ds \map {f_3} b\)

\(=\)

\(\ds v\)

\(\text {(4)}: \quad\)

\(\ds \map {f_4} a\)

\(=\)

\(\ds v\)

\(\ds \map {f_4} b\)

\(=\)

\(\ds u\)

$f_1$ and $f_4$ are bijections.

$f_2$ and $f_3$ are neither surjections nor injections.

$2$ Elements to $3$ Elements

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

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

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

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

a total of $3^2 = 9$.

All but the first, fifth and last are injections.

None are surjections.

$3$ 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.

Set of Cardinality $4$ to Itself

Let $S$ be a set whose cardinality is $4$:

$\card S = 4$

Then there are $256$ mappings from $S$ to itself.