Injective function: Difference between revisions

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== Injections may be made invertible ==

In fact, to turn an injective function <math>f : X \to Y</math> into a bijective (hence invertible) function, it suffices to replace its codomain <math>Y</math> by its actual rangeimage <math>J = f(X).</math> That is, let <math>g : X \to J</math> such that <math>g(x) = f(x)</math> for all <math>x \in X</math>; then <math>g</math> is bijective. Indeed, <math>f</math> can be factored as <math>\operatorname{In}_{J,Y} \circ g,</math> where <math>\operatorname{In}_{J,Y}</math> is the [[inclusion function]] from <math>J</math> into <math>Y.</math>

More generally, injective [[partial function]]s are called [[partial bijection]]s.