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

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## 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 $1 \ \text {(i)}$