Injective function

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,^[4]$${\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).}$$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.^[5]

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

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

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.^[2]

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 $${\displaystyle f(a)=f(b)\rightarrow g(f(a))=g(f(b))\rightarrow a=b.}$$

In this case, ${\displaystyle g}$ is called a retraction of ⁠${\displaystyle f}$⁠. Conversely, ${\displaystyle f}$ is called a section of ⁠${\displaystyle g}$⁠. For example: ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} ^{2},x\mapsto (1,m)^{\intercal }x}$ is retracted by ⁠${\displaystyle g:y\mapsto {\frac {(1,m)}{1+m^{2}}}y}$⁠.

Conversely, every injection ${\displaystyle f}$ with a non-empty domain has a left inverse ${\displaystyle g}$. It can be defined by choosing an element ${\displaystyle a}$ in the domain of ${\displaystyle f}$ and setting ${\displaystyle g(y)}$ to the unique element of the pre-image ${\displaystyle f^{-1}[y]}$ (if it is non-empty) or to ${\displaystyle a}$ (otherwise).^[6]

The left inverse ${\displaystyle g}$ is not necessarily an inverse of ${\displaystyle f,}$ because the composition in the other order, ⁠${\displaystyle f\circ g}$⁠, may differ from the identity on ⁠${\displaystyle Y}$⁠. 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.

In fact, to turn an injective function ${\displaystyle f:X\to Y}$ into a bijective (hence invertible) function, it suffices to replace its codomain ${\displaystyle Y}$ by its actual image ${\displaystyle J=f(X).}$ That is, let ${\displaystyle g:X\to J}$ such that ${\displaystyle g(x)=f(x)}$ for all ⁠${\displaystyle x\in X}$⁠; then ${\displaystyle g}$ is bijective. Indeed, ${\displaystyle f}$ can be factored as ⁠${\displaystyle \operatorname {In} _{J,Y}\circ g}$⁠, where ${\displaystyle \operatorname {In} _{J,Y}}$ is the inclusion function from ${\displaystyle J}$ into ⁠${\displaystyle Y}$⁠.

More generally, injective partial functions are called partial bijections.

• If ${\displaystyle f}$ and ${\displaystyle g}$ are both injective then ${\displaystyle f\circ g}$ is injective.
• If ${\displaystyle g\circ f}$ is injective, then ${\displaystyle f}$ is injective (but ${\displaystyle g}$ need not be).
• ${\displaystyle f:X\to Y}$ is injective if and only if, given any functions ⁠${\displaystyle g}$⁠, ${\displaystyle h:W\to X}$ whenever ⁠${\displaystyle f\circ g=f\circ h}$⁠, then ⁠${\displaystyle g=h}$⁠. In other words, injective functions are precisely the monomorphisms in the category Set of sets.
• If ${\displaystyle f:X\to Y}$ is injective and ${\displaystyle A}$ is a subset of ⁠${\displaystyle X}$⁠, then ⁠${\displaystyle f^{-1}(f(A))=A}$⁠. Thus, ${\displaystyle A}$ can be recovered from its image ⁠${\displaystyle f(A)}$⁠.
• If ${\displaystyle f:X\to Y}$ is injective and ${\displaystyle A}$ and ${\displaystyle B}$ are both subsets of ⁠${\displaystyle X}$⁠, then ⁠${\displaystyle f(A\cap B)=f(A)\cap f(B)}$⁠.
• Every function ${\displaystyle h:W\to Y}$ can be decomposed as ${\displaystyle h=f\circ g}$ for a suitable injection ${\displaystyle f}$ and surjection ⁠${\displaystyle g}$⁠. This decomposition is unique up to isomorphism, and ${\displaystyle f}$ may be thought of as the inclusion function of the range ${\displaystyle h(W)}$ of ${\displaystyle h}$ as a subset of the codomain ${\displaystyle Y}$ of ⁠${\displaystyle h}$⁠.
• If ${\displaystyle f:X\to Y}$ is an injective function, then ${\displaystyle Y}$ has at least as many elements as ${\displaystyle X,}$ in the sense of cardinal numbers. In particular, if, in addition, there is an injection from ⁠${\displaystyle Y}$⁠ to ⁠${\displaystyle X}$⁠, then ${\displaystyle X}$ and ${\displaystyle Y}$ have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
• If both ${\displaystyle X}$ and ${\displaystyle Y}$ are finite with the same number of elements, then ${\displaystyle f:X\to Y}$ is injective if and only if ${\displaystyle f}$ is surjective (in which case ${\displaystyle f}$ 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 ${\displaystyle f}$ is injective can be decided by only considering the graph (and not the codomain) of ⁠${\displaystyle f}$⁠.

A proof that a function ${\displaystyle f}$ 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 ⁠${\displaystyle f(x)=f(y)}$⁠, then ⁠${\displaystyle x=y}$⁠.^[7]

Here is an example: $${\displaystyle f(x)=2x+3}$$

Proof: Let ⁠${\displaystyle f:X\to Y}$⁠. Suppose ⁠${\displaystyle f(x)=f(y)}$⁠. So ${\displaystyle 2x+3=2y+3}$ implies ⁠${\displaystyle 2x=2y}$⁠, which implies ⁠${\displaystyle x=y}$⁠. Therefore, it follows from the definition that ${\displaystyle f}$ is injective.

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

• Bijection, injection and surjection – Properties of mathematical functions
• Injective metric space – Type of metric space
• Monotonic function – Order-preserving mathematical function
• Univalent function – Mathematical concept