# Injective function

A func­tion $$f: X \to Y$$ is in­jec­tive if it has the prop­erty that when­ever $$f(x) = f(y)$$, it is the case that $$x=y$$. Given an el­e­ment in the image, it came from ap­ply­ing $$f$$ to ex­actly one el­e­ment of the do­main.

This con­cept is also com­monly called be­ing “one-to-one”. That can be a lit­tle mis­lead­ing to some­one who does not already know the term, how­ever, be­cause many peo­ple’s nat­u­ral in­ter­pre­ta­tion of “one-to-one” (with­out oth­er­wise hav­ing learnt the term) is that ev­ery el­e­ment of the do­main is matched up in a one-to-one way with ev­ery el­e­ment of the do­main, rather than sim­ply with some el­e­ment of the do­main. That is, a rather nat­u­ral way of in­ter­pret­ing “one-to-one” is as “bi­jec­tive” rather than “in­jec­tive”.

# Examples

• The func­tion $$\mathbb{N} \to \mathbb{N}$$ (where $$\mathbb{N}$$ is the set of nat­u­ral num­bers) given by $$n \mapsto n+5$$ is in­jec­tive: since $$n+5 = m+5$$ im­plies $$n = m$$. Note that this func­tion is not sur­jec­tive: there is no nat­u­ral num­ber $$k$$ such that $$k+5 = 2$$, for in­stance, so $$2$$ is not in the range of the func­tion.

• The func­tion $$f: \mathbb{N} \to \mathbb{N}$$ given by $$f(n) = 6$$ for all $$n$$ is not in­jec­tive: since $$f(1) = f(2)$$ but $$1 \not = 2$$, for in­stance.

Parents:

• Why is it mis­lead­ing to call in­jec­tive “one-to-one”?

• I’ve ed­ited some­thing about that into the text. Ba­si­cally I think it’s to do with the sym­me­try of the words in “one-to-one”: it looks like it should go both ways, as “one thing in the do­main hits one thing in the range, and vice versa”.