# Cardinality of Set of All Mappings/Examples/3 Elements to 2 Elements

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## Example of Cardinality of Set of All Mappings

Let $S = \set {1, 2, 3}$.

Let $T = \set {a, b}$.

Then the mappings from $S$ to $T$ in two-row notation are:

- $\dbinom {1 \ 2 \ 3} {a \ a \ a}, \dbinom {1 \ 2 \ 3} {a \ a \ b}, \dbinom {1 \ 2 \ 3} {a \ b \ a}, \dbinom {1 \ 2 \ 3} {a \ b \ b}, \dbinom {1 \ 2 \ 3} {b \ a \ a}, \dbinom {1 \ 2 \ 3} {b \ a \ b}, \dbinom {1 \ 2 \ 3} {b \ b \ a}, \dbinom {1 \ 2 \ 3} {b \ b \ b}$

a total of $2^3 = 8$.

All but the first and last are surjections.

None are injections.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.2$. Equality of mappings - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Example $47$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $2$: Maps and relations on sets: Exercise $1$