Injection

Let be a function defined on a set and taking values in a set . Then is said to be an injection (or injective map, or embedding) if, whenever , it must be the case that . Equivalently, implies . In other words, is an injection if it maps distinct objects to distinct objects. An injection is sometimes also called one-to-one.

A linear transformation is injective if the kernel of the function is zero, i.e., a function is injective iff .

A function which is both an injection and a surjection is said to be a bijection.

In the categories of sets, groups, modules, etc., a monomorphism is the same as an injection, and is used synonymously with "injection" outside of category theory.

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