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

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 $1$