Subfunction 2009-01-05 16:06:09 2009-01-06 17:14:43 Definition partial function restriction \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} % define commands here \newcommand*{\abs}[1]{\left\lvert #1\right\rvert} \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} \textbf{Definition}. Let $f:A\to B$ and $g:C\to D$ be partial functions. $g$ is said to be a \emph{subfunction} of $f$ if $$g\subseteq f \cap (C\times D).$$ In other words, $g$ is a subfunction of $f$ iff whenever $x\in C$ such that $g(x)$ is defined, then $x\in A$, $f(x)$ is defined, and $g(x)=f(x)$. If we set $C'=A\cap C$ and $D'=B\cap D$, then $g\subseteq f\cap (C'\times D')$, so there is no harm in assuming that $C$ and $D$ are subsets of $A$ and $B$ respectively, which we will do for the rest of the discussion. In practice, whenever $g$ is a subfunction of $f$, we often assume that $g$ and $f$ have the same domain and codomain. Otherwise, we would specify that $g$ is a subfunction of $f:A\to B$ with domain $C$ and codomain $D$. For example, $f:\mathbb{R} \to \mathbb{R}$ defined by $$f(x)=\sqrt{x^2-1}$$ is a partial function, whose domain of definition is $(-\infty,-1]\cup [1,\infty)$, and the partial function $g:\mathbb{R} \to \mathbb{R}$ given by $$g(x)=\displaystyle{\frac{x^2-1}{\sqrt{x^2-1}}}$$ is a subfunction of $f$. The domain of definition of $g$ is $(-\infty,-1)\cup (1,\infty)$. Two immediate properties of a subfunction $g:C\to D$ of $f:A\to B$ are \begin{itemize} \item the range of $g$ is a subset of the range of $f$: $$g(C)\subseteq f(C),$$ \item the domain of definition of $g$ is a subset of the domain of definition of $f$: $$g^{-1}(D)\subseteq f^{-1}(D).$$ \end{itemize} \textbf{Definition}. A subfunction $g:C\to D$ of $f:A\to B$ is called a \emph{restriction of $f$ relative to $D$}, if $g(C)=f(C)\cap D$, and a \emph{restriction of $f$} if $g(C)=f(C)$. Every partial function $g:C\to D$ corresponds to a unique restriction $g':C\to g(C)$ of $g$. A restriction $g:C\to D$ of $f:A\to B$ is certainly a restriction of $f$ relative to $D$, since $f(C)\cap D = g(C)\cap D = g(C)$, but not conversely. For example, let $A$ be the set of all non-negative integers and $-_A: A^2\to A$ the ordinary subtraction. $-_A$ is easily seen to be a partial function. Let $B$ be the set of all positive integers. Then $-_B:B^2\to B$ is a restriction of $-_A:A^2\to A$, relative to $B$. However, $-_B$ is not a restriction of $-_A$, for $n -_B n$ is not defined, while $n -_A n = 0\in A$. \begin{thebibliography}{7} \bibitem{gg} G. Gr\"{a}tzer: {\em Universal Algebra}, 2nd Edition, Springer, New York (1978). \end{thebibliography}