Injective function

Written by Patrick Stevens last updated
Requires: Function

A function is injective if it has the property that whenever , it is the case that . Given an element in the image, it came from applying 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 (where is the set of natural numbers) given by is injective: since implies . Note that this function is not surjective: there is no natural number such that , for instance, so is not in the range of the function.
  • The function given by for all is not injective: since but , for instance.
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