# Cardinality of Set of Injections/Examples/Cardinality 2 to Cardinality 3

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## Examples of Cardinality of Set of Injections

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

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

Then the set of injections from $A$ to $B$, expressed in two-row notation, is:

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

thus demonstrating that there are $\dfrac {3!} {\paren {3 - 2}!} = \dfrac {3!} {1!} = \dfrac 6 1 = 6$ injections from $A$ to $B$.

## Proof

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $3$. Mappings: Exercise $1$