An injective function (injection)

Another injective function (this one is a bijection)

A non-injective function (this one happens to be a surjection)

In mathematics, an injective function is a function which associates distinct arguments with distinct values. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

An injective function is called an injection, and is also said to be an information-preserving or one-to-one function (the latter is not to be confused with one-to-one correspondence, i. e. a bijective function). In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

A function f that is not injective is sometimes called many-to-one. (However, this terminology is also sometimes used to mean "single-valued", i. e. each argument is mapped to at most one value. )

Definition

Let f be a function whose domain is a set A. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined It is injective if, for all a and b in A such that f(a)=f(b), we have a = b.

Examples and counter-examples

• For any set X, the identity function on X is injective. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that
• The function f : R → R defined by f(x) = 2x + 1 is injective.
• The function g : R → R defined by g(x) = x² is not injective, because (for example) g(1) = 1 = g(−1). However, if g is redefined so that its domain is the non-negative real numbers [0,+∞), then g is injective.
• The exponential function is injective (but not surjective as no value maps to a negative number). The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every
• The natural logarithm function is injective. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational
• The function g : R → R defined by g(x) = xⁿ − x is not injective, since, for example, g(0) = g(1).

More generally, when X and Y are both the real line R, then an injective function f : R → R is one whose graph is never intersected by any horizontal line more than once. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

Injections can be undone

Functions with left inverses are always injections. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B That is, given f : X → Y, if there is a function g : Y → X such that, for every

(f can be undone by g)

then f is injective. In this case, f is called a section of g and g is called a retraction of f. In Category theory, a branch of Mathematics, a section is a right inverse of a morphism In Category theory, a branch of Mathematics, a section is a right inverse of a morphism

Conversely, every injection f with non-empty domain has a left inverse g (in conventional mathematics^[1]). Note that g may not be a complete inverse of f because the composition in the other order, f o g, may not be the identity on Y. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In other words, a function that can be undone or "reversed", such as f, is not necessarily invertible (bijective). In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Injections are "reversible" but not always invertible.

Although it is impossible to reverse a non-injective (and therefore information-losing) function, you can at least obtain a "quasi-inverse" of it, that is a multiple-valued function. In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map

Injections may be made invertible

In fact, to turn an injective function f : X → Y into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range J = f(X). In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property That is, let g : X → J such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as incl_J,Yog, where incl_J,Yis the inclusion function from J into Y. In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the

Other properties

• If f and g are both injective, then f o g is injective.

The composition of two injective functions is injective.

• If g o f is injective, then f is injective (but g need not be).
• f : X → Y is injective if and only if, given any functions g, h : W → X, whenever f o g = f o h, then g = h. In other words, injective functions are precisely the monomorphisms in the category Set of sets. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are
• If f : X → Y is injective and A is a subset of X, then f ⁻¹(f(A)) = A. Thus, A can be recovered from its image f(A). In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage
• If f : X → Y is injective and A and B are both subsets of X, then f(A ∩ B) = f(A) ∩ f(B).
• Every function h : W → Y can be decomposed as h = f o g for a suitable injection f and surjection g. This decomposition is unique up to isomorphism, and f may be thought of as the inclusion function of the range h(W) of h as a subset of the codomain Y of h. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the
• If f : X → Y is an injective function, then Y has at least as many elements as X, in the sense of cardinal numbers. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.
• If both X and Y are finite with the same number of elements, then f : X → Y is injective if and only if f is surjective. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every
• Every embedding is injective. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group

See also

• surjective function
• injective module
• monomorphism
• Horizontal line test
• Injective metric space

Notes

1. ^ This principle is valid in conventional mathematics, but may fail in constructive mathematics. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, especially in the area of Abstract algebra known as Module theory, an injective module is a module Q that shares certain In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Mathematics, the horizontal line test is a test used to determine if a function is Injective, Surjective or Bijective. In Metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a Metric space with certain properties generalizing those In the Philosophy of mathematics For instance, a left inverse of the inclusion {0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set {0,1}. In Constructive mathematics, indecomposability or indivisibility (unzerlegbarkeit from the adjective unzerlegbar) is the principle that the In Category theory, a branch of Mathematics, a section is a right inverse of a morphism

References

• Bartle, Robert G. (1976), The Elements of Real Analysis (2nd ed. ), New York: John Wiley & Sons, ISBN 978-0-471-05464-1 , p. John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors 17 ff.
• Halmos, Paul R. (1974), Naive Set Theory, ISBN 978-0-387-90092-6 , p. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician See also Naive set theory for the mathematical topic Naive Set Theory is a Mathematics textbook by Paul Halmos 38 ff.