Injective function

 ( Functions And Mappings ) 

In mathematics, an injective function (also known as injection, or one-to-one functionSometimes one-one function, in Indian mathematical education. ) 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 , 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 $f$ that is not injective is sometimes called many-to-one.

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

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

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

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

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

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

$$f(a)\; =\; f(b)\; \backslash rightarrow\; g(f(a))=g(f(b))\; \backslash rightarrow\; a\; =\; b.$$

In this case, $g$ is called a retraction of $f.$ Conversely, $f$ is called a section of $g.$

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

The left inverse $g$ is not necessarily an Inverse function of $f,$ because the composition in the other order, $f\; \backslash circ\; g,$ may differ from the identity on $Y.$ In other words, an injective function can be "reversed" by a left inverse, but is not necessarily Inverse function, which requires that the function is bijective.

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

More generally, injective are called partial bijections.

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Other properties

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

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Proving that functions are injective A proof that a function $f$ 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 $f(x)\; =\; f(y),$ then $x\; =\; y.$

Here is an example: $$f(x)\; =\; 2\; x\; +\; 3$$

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

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

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Gallery

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See also

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Notes

• , p. 17 ff.
• , p. 38 ff.

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External links

• 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

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