Injection and Surjection
is injective when , and surjective when every satisfies for some .
Definition 2.15 (Injection, Surjection, Bijection)
We say is injective if , we have . Equivalently, is injective if by the contrapositive.
We say is surjective if , such that .
We say is bijective if it is both injective and surjective.
Observations on finite sets
is surjective if and only if . For finite sets and : if there is no surjective function from to ; if there is no injection from to ; and for with finite, is injective if and only if it is surjective.
Related
Stated in
- Definition 2.15 (Injection, Surjection, Bijection)§2.2 Functions