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{{Functions}}
{{hatnote|For visual examples, as well as [[
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.
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== Examples ==
''For visual examples, readers are directed to the [[
* For any set <math>X</math> and any subset <math>S \subseteq X,</math> the [[inclusion map]] <math>S \to X</math> (which sends any element <math>s \in S</math> to itself) is injective. In particular, the [[identity function]] <math>X \to X</math> 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.
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