1.5 Functions

1.57   Definition (Function.) Let $A,B$ be sets, and let $f$ be a rule that assigns to each element $a$ in $A$ a unique element (denoted by $f(a)$) in $B$. The ordered triple $(A,B,f)$ is called a function with domain $A$ and codomain $B$. We write

$$f\colon A\to B$$

to indicate that $(A,B,f)$ is a function. It follows from the definition that two functions are equal if and only if they have the same domain, the same codomain, and the same rule: If $f\colon A\to B$ and $g\colon A\to B$, I say that the rule $f$ and the rule $g$ are the same if and only if $f(a)=g(a)$ for all $a\in A$. We usually say `` the function $f$" when we mean `` the function $(A,B,f)$," i.e., we name a function by giving just the name for its rule.

1.58   Examples. Let

$$\begin{eqnarray*} &&f\colon\mbox{{\bf N}}\to\mbox{{\bf Z}}, \\ &&g\colon\mbox{{... ...\to\mbox{{\bf N}}, \\ &&k\colon\mbox{{\bf Z}}\to\mbox{{\bf Z}}, \end{eqnarray*}$$

be defined by

$$\begin{eqnarray*} f(n)&=&(n-2)(n-3) \mbox{ for all } n\in\mbox{{\bf N}}, \\ g(n... ...bf Z}}, \\ k(w)&=&w^2-5w+6 \mbox{ for all } w\in\mbox{{\bf Z}}. \end{eqnarray*}$$

Then

$$\begin{eqnarray*} f&\not=&g \quad (f \mbox{ and } g \mbox{ have different domain... ... (g \mbox{ and } h \mbox{ have different codomains }) \\ g&=&k. \end{eqnarray*}$$

If $P$ is a proposition form on a set $S$, then $P$ determines a function whose domain is $S$ and whose codomain is the set of all propositions.