Functions

Let $f$ be a function defined on a set $A$ and taking values in a set $B$. Set $A$ is called the domain of $f$; set $B$ is called the range of $f$. Function $f$ is said to be one-to-one (i.e., an injection or embedding) if, whenever $f(x)= f(y)$, it must be the case that $x=y$. In other words, $f$ is one-to-one if it maps distinct objects to distinct objects. Function $f$ is said to be onto (a.k.a. a surjection) if, for any $b \in B$, there exists an $a \in A$ for which $f(a) = b$. A function is bijection if it is one to one (i.e., injection) and onto (i.e., surjection). If function $f$ is defined for each element of $A$, then $f$ is said to be a total function; otherwise, $f$ is said to be a partial function.