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”.


  • 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.