empty product EmptyProduct 2004-11-08 18:25:35 2008-05-22 12:35:05 Definition unity % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % 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} % there are many more packages, add them here as you need them % define commands here The {\em empty product} of numbers is the borderline case of product, where the number of \PMlinkescapetext{factors is zero, i.e. the set of the factors} is empty. \,The most usual examples are the following. \begin{itemize} \item The \PMlinkname{zeroth power}{Introducing0thPower} of a non-zero number:\, $a^0$ \item The factorial of 0:\, 0! \item The \PMlinkname{prime factor presentation}{FundamentalTheoremOfArithmetics} of unity, which has no prime factors \end{itemize} The value of the empty sum of numbers is equal to the additive identity number, 0.\, Similarly, the empty product of numbers is equal to the \PMlinkname{multiplicative identity}{Unity} number, 1. \textbf{Note.}\, When considering the complex numbers as pairs of real numbers one often identifies the pairs $(x,\,0)$ and the reals $x$.\, In this sense one can think that the Cartesian product $\mathbb{R}\times\{0\}$ is equal to $\mathbb{R}$.\, This seems to \PMlinkescapetext{mean} the equation $$\mathbb{R}\times\mathbb{R}^0 = \mathbb{R}^{1+0} = \mathbb{R}^1 = \mathbb{R},$$ although the \PMlinkname{associativity}{GeneralAssociativity} of Cartesian product is nowhere stated.\, Nevertheless, it is sometimes natural to define that the Cartesian product of an empty collection of sets equals to a set with one element; so it may \PMlinkescapetext{mean} that e.g.\, $\mathbb{R}^0 = \{0\}.$ One can also consider empty products in categories. It follows directly from the definition that an object in a category is a \PMlinkname{product}{CategoricalDirectProduct} of an empty family of objects in the category if and only if it is a terminal object of the category. Sets are a special case of this: in the category of sets the singletons are the terminal objects, so the empty product exists and is a singleton.