Set 2025-11-22 22:09:22 2025-11-23 01:09:33 Definition % this is the default PhysicsLibrary 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 \section{Set} \subsection{Definition} In mathematics, a set is a collection of different things; the things are elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets themselves are purely conceptual. This is an important point to note: the set of all cows (for example) does not physically exist, even though the cows do. The set is a "gathering" of the cows into one conceptual unit that is not part of physical reality. This makes it easy to see why we can have sets with an infinite number of elements; even though we may not be able to point out infinitely many objects in the real world, we can construct conceptual sets which an infinite number of elements. \\ Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo-Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. \subsection{Basic Notation} If $x$ is an element of a set $S$, one says that $x$ \emph{belongs} to $S$ or \emph{is in} $S$, and this is written as $x \,\, \epsilon \,\, S$ [11]. The statement "$y$ is not in $S$" is written as $y \, \notin \, S$ , which can also be read as "$y$ is not in S"[12][13]. For example, if $\mathbb{Z}$ is the set of the integers, one has $-3 \,\, \epsilon \,\, \mathbb{Z}$ and $1.5 \, \notin \, \mathbb{Z}$. Each set is uniquely characterized by its elements. $$ A = B \iff \forall x \left( x \,\, \epsilon \,\, A \iff x \,\, \epsilon \,\, B \right). $$ This implies that there is only one set with no element, the empty set (or null set) that is denoted $\varnothing$, {$\emptyset$} or $\{\}$[17][18]. A singleton is a set with exactly one element. If $x$ is this element, the singleton is denoted $\{x\}$.