Cardinality of Set of Bijections/Examples/Set of Cardinality 4
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Examples of Use of Cardinality of Set of Bijections
Let $S$ be a set whose cardinality is $4$:
- $\card S = 4$
Then there are $24$ bijections from $S$ to itself.
Proof
Let $T$ be the set of bijections from $S$ to itself.
From Cardinality of Set of Bijections:
- $\card T = \paren {\card S}!$
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 $1 \ \text {(ii)}$