Editing Injective function (section)

== Definition ==
[[file:Injection.svg|thumb|An injective function, which is not also [[Surjective function|surjective]].]]
{{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=Section 4.2 Injections, Surjections, and Bijections |last=Farlow|first=S. J.|author-link= Stanley Farlow |website=Mathematics & 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>An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, <math>f:A\rightarrowtail B</math> or  <math>f:A\hookrightarrow B</math>), although some authors specifically reserve ↪ for an [[inclusion map]].<ref>{{Cite web |title=What are usual notations for surjective, injective and bijective functions? |url=https://math.stackexchange.com/questions/46678/what-are-usual-notations-for-surjective-injective-and-bijective-functions |access-date=2024-11-24 |website=Mathematics Stack Exchange |language=en}}</ref>