Injective, surjective and bijective functions

Three properties classify functions according to how they “cover” the codomain.

Definition — Injective, surjective, bijective

A function $f:A\to B$ is said to be:

• injective if $x_1\ne x_2 \implies f(x_1) \ne f(x_2)$, that is, if distinct elements of the domain correspond to distinct images;
• surjective if $\mathrm{Im}(f) = B$, that is, if every element of the codomain is the image of at least one element of the domain;
• bijective (or one-to-one and onto) if it is both injective and surjective.

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Topics: Functions and properties
Concepts: Bijective function · Injective function · Surjective function · Image