Injective function: Difference between revisions

Revision as of 16:30, 5 September 2022

Revision as of 16:49, 5 September 2022

Line 6:

{{hatnote|For visual examples, as well as [[mathematical intuition]], readers are directed to the [[#Gallery|gallery section.]]}}

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 of its ___domain to distinct elements; that is, {{math|''f''(''x''₁) {{=}} ''f''(''x''₂)}} implies {{math|1=''x''₁ = ''x''₂}} for any two elements {{math|''x''₁}} and {{math|''x''₁}} of {{math|''f''}}'s ___domain. (Equivalently, {{math|1=''x''₁ ≠ ''x''₂}} implies {{math|''f''(''x''₁) {{≠}} ''f''(''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.