Cardinality of Set of Self-Mappings on Finite Set
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Theorem
Let $S$ be a finite set.
Let the cardinality of $S$ be $n$.
The cardinality of the set of all mappings from $S$ to itself (that is, the total number of self-maps on $S$) is:
- $\card {S^S} = n^n$
Proof
This is a specific example of Cardinality of Set of All Mappings where $S = T$.
$\blacksquare$
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$: Definition and examples of semigroups