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

Jump to navigation
Jump to search

## 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 $1$