injection

An injection, or an injective function, is a function $f:A\to B$ such that no two arguments have the same value, i.e. each element of the image is only used once. For each $b$ in $f[A]$, there exists only one ordered pair $⟨a,b⟩$ for any $a\in A$. For example, the function $f(x):\mathbb{N}\to \mathbb{N}$ such that $f(x)=x^2$is injective, since each natural number can only be the square of at most one natural number. $g:\mathbb{Z}\to \mathbb{N}$ such that $g(x)=x^2$ is not injective, since $g(x)=g(-x)$.