Injective function

In mathematics, an injective function (also known as injection, or one-to-one function^[1] ) is a function $f$ that maps distinct elements of its domain to distinct elements; that is, $x 1 \neq x 2$ implies $f (x 1) \neq f (x 2)$ . (Equivalently, $f (x 1) = f (x 2)$ implies $x 1 = x 2$ in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of at most one element of its domain.^[2] The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism.^[3] This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.

A function $f$ that is not injective is sometimes called many-to-one.^[2]

^ Sometimes one-one function, in Indian mathematical education. "Chapter 1:Relations and functions" (PDF). Archived (PDF) from the original on Dec 26, 2023 – via NCERT.
^ ^a ^b "Injective, Surjective and Bijective". Math is Fun. Retrieved 2019-12-07.
^ "Section 7.3 (00V5): Injective and surjective maps of presheaves". The Stacks project. Retrieved 2019-12-07.

[1] Sometimes one-one function, in Indian mathematical education. "Chapter 1:Relations and functions" (PDF). Archived (PDF) from the original on Dec 26, 2023 – via NCERT.

[:0-2] "Injective, Surjective and Bijective". Math is Fun. Retrieved 2019-12-07.

[3] "Section 7.3 (00V5): Injective and surjective maps of presheaves". The Stacks project. Retrieved 2019-12-07.

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Function
$x \mapsto f (x)$
History of the function concept
Examples of domains and codomains
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Binary relation Partial Multivalued Implicit Space
v t e