1.2: Functions

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If \(A\) and \(B\) are sets, we call a relation \(R \subset A \times B\) a function with domain \(A\) if for every \(a \in A\) there exists one, and only one, \(b \in B\) such that \((a, b) \in R .\) We typically indicate such a relation with the notation \(f: A \rightarrow B,\) and write \(f(a)=b\) to indicate that \((a, b) \in R .\) We call the set of all \(b \in B\) such that \(f(a)=b\) for some \(a \in A\) the range of \(f .\) With this notation, we often refer to \(R\) as the graph of \(f\).

We say \(f: A \rightarrow B\) is one-to-one if for every \(b\) in the range of \(f\) there exists a unique \(a \in A\) such that \(f(a)=b .\) We say \(f\) is onto if for every \(b \in B\) there exists at least one \(a \in A\) such that \(f(a)=b .\) For example, the function \(f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}\) defined by \(f(z)=z^{2}\) is one-to-one, but not onto, whereas the function \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) defined by \(f(z)=z+1\) is both one-to-one and onto.

Given two functions, \(g: A \rightarrow B\) and \(f: B \rightarrow C,\) we define the composition, denoted \(f \circ g: A \rightarrow C,\) to be the function defined by \(f \circ g(a)=f(g(a))\).

If \(f: A \rightarrow B\) is both one-to-one and onto, then we may define a function \(f^{-1}: B \rightarrow A\) by requiring \(f^{-1}(b)=a\) if and only if \(f(a)=b\). Note that this implies that \(f \circ f^{-1}(b)=b\) for all \(b \in B\) and \(f^{-1} \circ f(a)=a\) for all \(a \in A .\) We call \(f^{-1}\) the inverse of \(f\).

Given any collection of nonempty sets, \(\left\{A_{\alpha}\right\}, \alpha \in I,\) we assume the existence of a function \(\phi: I \rightarrow B=\bigcup_{\alpha \in I} A_{\alpha},\) with the property that \(\phi(\alpha) \in A_{\alpha} .\) We call such a function a choice function. The assumption that choice functions always exist is known as the Axiom of Choice.