In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with 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 is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see ' for more details.

A function <math>f</math> that is not injective is sometimes called many-to-one.<math display="block">\forall a, b \in X, \;\; a \neq b \Rightarrow f(a) \neq f(b).</math>An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, <math>f:A\rightarrowtail B</math> or ), although some authors specifically reserve ↪ for an inclusion map.

Examples

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

  • For any set <math>X</math> and any subset , the inclusion map <math>S \to X</math> (which sends any element <math>s \in S</math> to itself) is injective. In particular, the identity function <math>X \to X</math> 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 <math>f : \R \to \R</math> defined by <math>f(x) = 2 x + 1</math> is injective.
  • The function <math>g : \R \to \R</math> defined by <math>g(x) = x^2</math> is injective, because (for example) <math>g(1) = 1 = g(-1).</math> However, if <math>g</math> is redefined so that its domain is the non-negative real numbers , then <math>g</math> is injective.
  • The exponential function <math>\exp : \R \to \R</math> defined by <math>\exp(x) = e^x</math> is injective (but not surjective, as no real value maps to a negative number).
  • The natural logarithm function <math>\ln : (0, \infty) \to \R</math> defined by <math>x \mapsto \ln x</math> is injective.
  • The function <math>g : \R \to \R</math> defined by <math>g(x) = x^n - x</math> is not injective, since, for example, .

More generally, when <math>X</math> and <math>Y</math> are both the real line , then an injective function <math>f : \R \to \R</math> is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the .

Here is an example:

<math display="block">f(x) = 2 x + 3</math>

Proof: Let . Suppose . So <math>2 x + 3 = 2 y + 3</math> implies , which implies . Therefore, it follows from the definition that <math>f</math> is injective.

There are multiple other methods of proving that a function is injective. For example, in calculus if <math>f</math> 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 <math>f</math> is a linear transformation it is sufficient to show that the kernel of <math>f</math> contains only the zero vector. If <math>f</math> 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 <math>f</math> of a real variable <math>x</math> is the horizontal line test. If every horizontal line intersects the curve of <math>f(x)</math> in at most one point, then <math>f</math> is injective or one-to-one.

See also

Notes

References

  • , p.&nbsp;17 ff.
  • , p.&nbsp;38 ff.
  • 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