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.

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