Injective function information

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.

Injective function information

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Injective function

Surjective function

Partial function

Embedding

Inverse function

Identity function

Countable set

Pigeonhole principle

Twelvefold way

Univalent function

Injection

Bijection

Perfect hash function

Monomorphism

Inverse function theorem

Cardinality

Integration by substitution

Simple path

Finite set

Local diffeomorphism

Uncountable set

Function composition

Inclusion map

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
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