# 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”.