1.3.8: Summary
A function \(f\colon A \to B\) maps every element of the domain \(A\) to a unique element of the codomain \(B\) . If \(x \in A\) , we call the \(y\) that \(f\) maps \(x\) to the value \(f(x)\) of \(f\) for argument \(x\) . If \(A\) is a set of pairs, we can think of the function \(f\) as taking two arguments. The range \(\ran{f}\) of \(f\) is the subset of \(B\) that consists of all the values of \(f\) .
If \(\ran{f} = B\) then \(f\) is called surjective . The value \(f(x)\) is unique in that \(f\) maps \(x\) to only one \(f(x)\) , never more than one. If \(f(x)\) is also unique in the sense that no two different arguments are mapped to the same value, \(f\) is called injective . Functions which are both injective and surjective are called bijective .
Bijective functions have a unique inverse function \(f^{-1}\) . Functions can also be chained together: the function \((g \circ f)\) is the composition of \(f\) with \(g\) . Compositions of injective functions are injective, and of surjective functions are surjective, and \((f^{-1} \circ f)\) is the identity function.
If we relax the requirement that \(f\) must have a value for every \(x \in A\) , we get the notion of a partial functions . If \(f\colon A \pto B\) is partial, we say \(f(x)\) is defined , \(f(x) \fdefined\) if \(f\) has a value for argument \(x\) , and otherwise we say that \(f(x)\) is undefined , \(f(x) \fundefined\) . Any (partial) function \(f\) is associated with the graph \(R_f\) of \(f\) , the relation that holds iff \(f(x) = y\) .