Injective function: Difference between revisions

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{{Functions}}

In [[mathematics]], an '''injective function''' (also known as '''injection''', or '''one-to-one function'''<ref>Sometimes ''one-one function'', in Indian mathematical education. {{Cite web |title=Chapter 1:Relations and functions |url=https://ncert.nic.in/ncerts/l/lemh101.pdf |via=NCERT |url-status=live |archive-url=https://web.archive.org/web/20231226194119/https://ncert.nic.in/ncerts/l/lemh101.pdf |archive-date= Dec 26, 2023 }}</ref> ) is a [[function (mathematics)|function]] {{math|''f''}} that maps [[Distinct (mathematics)|distinct]] elements of its ___domain to distinct elements; that is, {{math|1=''x''₁ ≠ ''x''₂}} implies {{math|''f''(''x''₁) {{≠}} ''f''(''x''₂)}}. (Equivalently, {{math|''f''(''x''₁) {{=}} ''f''(''x''₂)}} implies {{math|1=''x''₁ = ''x''₂}} 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.comMath is Fun |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.

In [[mathematics]], an '''injective function''' (also known as '''injection''', or '''one-to-one function'''<ref>Sometimes ''one-one function'', in Indian mathematical education.

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—Thepresheaves Stacks project|website=stacks.math.columbia.eduThe Stacks project |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.

{{Cite web |title=Chapter 1:Relations and functions |url=https://ncert.nic.in/ncerts/l/lemh101.pdf}}</ref> ) is a [[function (mathematics)|function]] {{math|''f''}} that maps [[Distinct (mathematics)|distinct]] elements of its ___domain to distinct elements; that is, {{math|1=''x''₁ ≠ ''x''₂}} implies {{math|''f''(''x''₁) {{≠}} ''f''(''x''₂)}}. (Equivalently, {{math|''f''(''x''₁) {{=}} ''f''(''x''₂)}} implies {{math|1=''x''₁ = ''x''₂}} 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" />

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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=Section 4.2 Injections, Surjections, and Bijections |last=Farlow|first=S. J.|author-link= Stanley Farlow |website=math.umaine.eduMathematics & Statistics - University of Maine |access-date=2019-12-06 |url-status=dead |archive-url= https://web.archive.org/web/20191207035302/http://www.math.umaine.edu/~farlow/sec42.pdf |archive-date= Dec 7, 2019 }}</ref><math display="block">\forall a, b \in X, \;\; a \neq b \Rightarrow f(a) \neq f(b).</math>

== Examples ==

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A proof that a function <math>f</math> 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 <math>f(x) = f(y),</math> then <math>x = y.</math><ref>{{cite web|last=Williams|first=Peter|title=Proving Functions One-to-One|url=http://www.math.csusb.edu/notes/proofs/bpf/node4.html |date=Aug 21, 1996 |website=Department of Mathematics at CSU San Bernardino Reference Notes Page |archive-date= 4 June 2017|archive-url=https://web.archive.org/web/20170604162511/http://www.math.csusb.edu/notes/proofs/bpf/node4.html}}</ref>

Here is an example: