What does injective mean?
Injective is the standard mathematical term for a function or mapping that sends different inputs to different outputs. Such a function is one-to-one: if two inputs have the same output, they must be the same input. Injectivity is an important property because it preserves distinctions, and the existence of an injective map from one structure to another can establish that the structures have certain corresponding properties. The term also appears in category theory, where an injective morphism has no two distinct objects mapped to the same object. In ordinary conversation, injective is rare because its idea is usually expressed more simply as one-to-one. It should not be confused with surjective, a complementary property meaning that every element of the target set is reached. Knowing both terms helps distinguish whether a mapping preserves distinctness, reaches every possible target, or does both.
adjectiveOf or relating to an injective function or mapping: one that assigns different outputs to different inputs. In category theory, an injective morphism likewise has no two distinct elements mapped to the same object.
- Relating to a function or mapping that assigns distinct outputs to distinct inputs; in category theory, relating to a morphism with no two distinct objects having the same image.
"The homomorphism between these two groups is injective, so distinct elements remain distinct under the map."
"An injective function preserves the distinction between every pair of different elements in its domain."
"The proof establishes that the proposed mapping is injective and therefore that it has a left inverse."
Literally derived from Latin for “thrown in,” injective describes a mapping that never sends two different inputs to the same destination.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Where does injective come from?
The word comes from Latin injectivus, derived from injectus, the past participle of injicere, meaning “to throw in” or “to drive in.” Latin injicere combines the prefix in- with jacere, “to throw,” the same verbal root found in words such as object, reject, and interject. English initially used the term in mathematical senses before it also became established in category theory.
Related word forms
How injective is actually used
This is primarily specialist mathematical terminology, especially common in algebra, topology, and category theory. Outside technical writing, it is uncommon and may be encountered in discussions of injections, injectivity, and one-to-one correspondences.
Easily confused with injective
An injective function maps distinct inputs to distinct outputs, whereas a surjective function maps at least one input to every output in the target set.