Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x₁ ≠ x₂ implies f(x₁) ≠ f(x₂) (equivalently by contraposition, f(x₁) = f(x₂) implies x₁ = x₂). In other words, every element of the function's codomain is the image of at most one element of its domain. 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. This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.

A function ${\displaystyle f}$ that is not injective is sometimes called many-to-one.

Let ${\displaystyle f}$ be a function whose domain is a set ${\displaystyle X.}$ The function ${\displaystyle f}$ is said to be injective provided that for all ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle X,}$ if ${\displaystyle f(a)=f(b),}$ then ${\displaystyle a=b}$; that is, ${\displaystyle f(a)=f(b)}$ implies ${\displaystyle a=b.}$ Equivalently, if ${\displaystyle a\neq b,}$ then ${\displaystyle f(a)\neq f(b)}$ in the contrapositive statement.

Symbolically,$${\displaystyle \forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,}$$ which is logically equivalent to the contrapositive,$${\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).}$$An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, ${\displaystyle f:A\rightarrowtail B}$ or ${\displaystyle f:A\hookrightarrow B}$), although some authors specifically reserve ↪ for an inclusion map.

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

More generally, when ${\displaystyle X}$ and ${\displaystyle Y}$ are both the real line ${\displaystyle \mathbb {R} ,}$ then an injective function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the horizontal line test.

Functions with left inverses are always injections. That is, given ${\displaystyle f:X\to Y,}$ if there is a function ${\displaystyle g:Y\to X}$ such that for every ${\displaystyle x\in X}$, ${\displaystyle g(f(x))=x}$, then ${\displaystyle f}$ is injective. The proof is that