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. In this case, ${\displaystyle g}$  is called a retraction of ${\displaystyle f.}$  Conversely, ${\displaystyle f}$  is called a section of ${\displaystyle g.}$ 

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

• Bartle, Robert G. (1976), The Elements of Real Analysis (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-05464-1, p. 17 ff.
• Halmos, Paul R. (1974), Naive Set Theory, New York: Springer, ISBN 978-0-387-90092-6, p. 38 ff.

 

Wikimedia Commons has media related to Injectivity.

 

Look up injective in Wiktionary, the free dictionary.

• 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