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, x1x2 implies f(x1) ≠ f(x2). (Equivalently, f(x1) = f(x2) implies x1 = x2 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 that is not injective is sometimes called many-to-one.[2]

Definition edit

An injective function, which is not also surjective

Let   be a function whose domain is a set   The function   is said to be injective provided that for all   and   in   if   then  ; that is,   implies   Equivalently, if   then   in the contrapositive statement.


which is logically equivalent to the contrapositive,[4]

Examples edit

For visual examples, readers are directed to the gallery section.

  • For any set   and any subset   the inclusion map   (which sends any element   to itself) is injective. In particular, the identity function   is always injective (and in fact bijective).
  • If the domain of a function is the empty set, then the function is the empty function, which is injective.
  • If the domain of a function has one element (that is, it is a singleton set), then the function is always injective.
  • The function   defined by   is injective.
  • The function   defined by   is not injective, because (for example)   However, if   is redefined so that its domain is the non-negative real numbers [0,+∞), then   is injective.
  • The exponential function   defined by   is injective (but not surjective, as no real value maps to a negative number).
  • The natural logarithm function   defined by   is injective.
  • The function   defined by   is not injective, since, for example,  

More generally, when   and   are both the real line   then an injective function   is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the horizontal line test.[2]

Injections can be undone edit

Functions with left inverses are always injections. That is, given   if there is a function   such that for every  ,  , then   is injective. In this case,   is called a retraction of   Conversely,   is called a section of  

Conversely, every injection   with a non-empty domain has a left inverse  . It can be defined by choosing an element   in the domain of   and setting   to the unique element of the pre-image   (if it is non-empty) or to   (otherwise).[5]

The left inverse   is not necessarily an inverse of   because the composition in the other order,   may differ from the identity on   In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.

Injections may be made invertible edit

In fact, to turn an injective function   into a bijective (hence invertible) function, it suffices to replace its codomain   by its actual range   That is, let   such that   for all  ; then   is bijective. Indeed,   can be factored as   where   is the inclusion function from   into  

More generally, injective partial functions are called partial bijections.

Other properties edit

The composition of two injective functions is injective.
  • If   and   are both injective then   is injective.
  • If   is injective, then   is injective (but   need not be).
  •   is injective if and only if, given any functions     whenever   then   In other words, injective functions are precisely the monomorphisms in the category Set of sets.
  • If   is injective and   is a subset of   then   Thus,   can be recovered from its image  
  • If   is injective and   and   are both subsets of   then  
  • Every function   can be decomposed as   for a suitable injection   and surjection   This decomposition is unique up to isomorphism, and   may be thought of as the inclusion function of the range   of   as a subset of the codomain   of  
  • If   is an injective function, then   has at least as many elements as   in the sense of cardinal numbers. In particular, if, in addition, there is an injection from   to   then   and   have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
  • If both   and   are finite with the same number of elements, then   is injective if and only if   is surjective (in which case   is bijective).
  • An injective function which is a homomorphism between two algebraic structures is an embedding.
  • Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function   is injective can be decided by only considering the graph (and not the codomain) of  

Proving that functions are injective edit

A proof that a function   is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if   then  [6]

Here is an example:


Proof: Let   Suppose   So   implies   which implies   Therefore, it follows from the definition that   is injective.

There are multiple other methods of proving that a function is injective. For example, in calculus if   is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if   is a linear transformation it is sufficient to show that the kernel of   contains only the zero vector. If   is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.

A graphical approach for a real-valued function   of a real variable   is the horizontal line test. If every horizontal line intersects the curve of   in at most one point, then   is injective or one-to-one.

Gallery edit

See also edit

Notes edit

  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.
  2. ^ a b c "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.
  4. ^ Farlow, S. J. "Section 4.2 Injections, Surjections, and Bijections" (PDF). Mathematics & Statistics - University of Maine. Archived from the original (PDF) on Dec 7, 2019. Retrieved 2019-12-06.
  5. ^ Unlike the corresponding statement that every surjective function has a right inverse, this does not require the axiom of choice, as the existence of   is implied by the non-emptiness of the domain. However, this statement may fail in less conventional mathematics such as constructive mathematics. In constructive mathematics, the inclusion   of the two-element set in the reals cannot have a left inverse, as it would violate indecomposability, by giving a retraction of the real line to the set {0,1}.
  6. ^ Williams, Peter (Aug 21, 1996). "Proving Functions One-to-One". Department of Mathematics at CSU San Bernardino Reference Notes Page. Archived from the original on 4 June 2017.

References edit

External links edit

  • Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.
  • Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injective functions