Cardinality of Set of All Mappings/Examples/2 Elements to 3 Elements
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Example of Cardinality of Set of All Mappings
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.
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Exercise $1$