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{{functions}}In [[mathematics]], an '''injective function''' (also known as '''injection''', or '''one-to-one function''') is a [[function (mathematics)|function]] {{math|''f''}} that maps [[Distinct (mathematics)|distinct]] elements to distinct elements; that is, {{math|''f''(''x''<sub>1</sub>) {{=}} ''f''(''x''<sub>2</sub>)}} implies {{math|1=''x''<sub>1</sub> = ''x''<sub>2</sub>}}. (Equivalently, {{math|1=''x''<sub>1</sub> ≠ ''x''<sub>2</sub>}} implies {{math|''f''(''x''<sub>1</sub>) {{≠}} ''f''(''x''<sub>2</sub>)}} in the equivalent [[Contraposition|contrapositive]] statement.) In other words, every element of the function's [[codomain]] is the [[Image (mathematics)|image]] of {{em|at most}} one element of its [[Domain of a function|domain]].<ref name=":0">{{Cite web|url=https://www.mathsisfun.com/sets/injective-surjective-bijective.html|title=Injective, Surjective and Bijective|website=www.mathsisfun.com|access-date=2019-12-07}}</ref> The term {{em|one-to-one function}} must not be confused with {{em|one-to-one correspondence}} that refers to [[bijective function]]s, which are functions such that each element in the codomain is an image of exactly one element in the domain.
A [[homomorphism]] between [[algebraic structure]]s is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for [[vector space]]s, an {{em|injective homomorphism}} is also called a {{em|[[monomorphism]]}}. However, in the more general context of [[category theory]], the definition of a monomorphism differs from that of an injective homomorphism.<ref>{{Cite web|url=https://stacks.math.columbia.edu/tag/00V5|title=Section 7.3 (00V5): Injective and surjective maps of presheaves—The Stacks project|website=stacks.math.columbia.edu|access-date=2019-12-07}}</ref> This is thus a theorem that they are equivalent for algebraic structures; see {{slink|Homomorphism|Monomorphism}} for more details.
A function <math>f</math> that is not injective is sometimes called many-to-one.<ref name=":0" />
== Definition ==
{{Further|topic=notation|Function (mathematics)#Notation}}
Let <math>f</math> be a function whose domain is a set <math>X.</math> The function <math>f</math> is said to be '''injective''' provided that for all <math>a</math> and <math>b</math> in <math>X,</math> if <math>f(a) = f(b),</math> then <math>a = b</math>; that is, <math>f(a) = f(b)</math> implies <math>a=b.</math> Equivalently, if <math>a \neq b,</math> then <math>f(a) \neq f(b)</math> in the [[Contraposition|contrapositive]] statement.
Symbolically,<math display="block">\forall a,b \in X, \;\; f(a)=f(b) \Rightarrow a=b,</math>
which is logically equivalent to the [[Contraposition|contrapositive]],<ref>{{Cite web|url=http://www.math.umaine.edu/~farlow/sec42.pdf|title=Injections, Surjections, and Bijections|last=Farlow|first=S. J.|author-link= Stanley Farlow |website=math.umaine.edu|access-date=2019-12-06}}</ref><math display="block">\forall a, b \in X, \;\; a \neq b \Rightarrow f(a) \neq f(b).</math>
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