Injective function: Difference between revisions

Revision as of 07:48, 3 July 2025

Revision as of 16:57, 7 August 2025

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== Definition ==

{{Dark mode invert|[[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.

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== Other properties ==

{{See also|List of set identities and relations#Functions and sets}}

{{Dark mode invert|[[Image:Injective composition2.svg|thumb|300px|The composition of two injective functions is injective.]]}}

* If <math>f</math> and <math>g</math> are both injective then <math>f \circ g</math> is injective.

* If <math>g \circ f</math> is injective, then <math>f</math> is injective (but <math>g</math> need not be).