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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

\begin{displaymath}f\colon A\to B\end{displaymath}

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.


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Next: 1.6 Russell's Paradox Up: 1. Notation, Undefined Concepts Previous: 1.4 More Sets   Index