Cardinality of Set of All Mappings/Examples/Set of Cardinality 4

Example of Cardinality of Set of All Mappings

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

$\card S = 4$

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

Proof

Let $T$ be the set of mappings from $S$ to itself.

From Cardinality of Set of All Mappings:

$\card T = \card S^\card S = 4^4 = 256$

The result follows by Examples of Factorials.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings: Exercise $1 \ \text {(i)}$