Editing Injective function (section)

== Other properties ==
{{See also|List of set identities and relations#Functions and sets}}
[[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).
* <math>f : X \to Y</math> is injective if and only if, given any functions <math>g,</math> <math>h : W \to X</math> whenever <math>f \circ g = f \circ h,</math> then <math>g = h.</math> In other words, injective functions are precisely the [[monomorphism]]s in the [[category theory|category]] '''[[Category of sets|Set]]''' of sets.
* If <math>f : X \to Y</math> is injective and <math>A</math> is a [[subset]] of <math>X,</math> then <math>f^{-1}(f(A)) = A.</math> Thus, <math>A</math> can be recovered from its [[Image (function)|image]] <math>f(A).</math>
* If <math>f : X \to Y</math> is injective and <math>A</math> and <math>B</math> are both subsets of <math>X,</math> then <math>f(A \cap B) = f(A) \cap f(B).</math>
* Every function <math>h : W \to Y</math> can be decomposed as <math>h = f \circ g</math> for a suitable injection <math>f</math> and surjection <math>g.</math> This decomposition is unique [[up to isomorphism]], and <math>f</math> may be thought of as the [[inclusion function]] of the range <math>h(W)</math> of <math>h</math> as a subset of the codomain <math>Y</math> of <math>h.</math>
* If <math>f : X \to Y</math> is an injective function, then <math>Y</math> has at least as many elements as <math>X,</math> in the sense of [[cardinal number]]s. In particular, if, in addition, there is an injection from <math>Y</math> to <math>X,</math> then <math>X</math> and <math>Y</math> have the same cardinal number. (This is known as the [[Cantor–Bernstein–Schroeder theorem]].)
* If both <math>X</math> and <math>Y</math> are [[Finite set|finite]] with the same number of elements, then <math>f : X \to Y</math> is injective if and only if <math>f</math> is surjective (in which case <math>f</math> is bijective).
* An injective function which is a homomorphism between two algebraic structures is an [[embedding]].
* Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function <math>f</math> is injective can be decided by only considering the graph (and not the codomain) of <math>f.</math>