1.2: Functions

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.