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documentclass{article} %%% remove comment delimiter (´%´) and ... \documentclass{article} %%% remove comment delimiter ('%') and specify encoding parameter if required, %%% see TeX documentation for additional info (cp1252-Western,cp1251-Cyrillic) %\usepackage[cp1252]{inputenc} %%% remove comment delimiter ('%') and select language if required %\usepackage[english,spanish]{babel} \usepackage{amssymb} \usepackage{amsmath} \usepackage[dvips]{graphicx} %%% remove comment delimiter ('%') and specify parameters if required %\usepackage[dvips]{graphics} \begin{document} %%% remove comment delimiter ('%') and select language if required %\selectlanguage{spanish} \textbf{Binary Relations and Equivalence Relations and Partitions} \textbf{Relations: } Relation, a mathematical concept, is a set of ordered pairs. More precisely, the concept is called \textbf{The Definition of Relation} Let X and Y be sets. A relation, R, from X to Y is a subset of the Cartesian product X $\times$ Y. Let x be an element of X and y be an element of Y. The notations (x, y) is an element of R and x R If X = Y, then R is called a relation on X and is, of course, a subset of X $\times$ X. \textbf{Example of Relation} Let X and Y be sets. The trivial relation is the empty set, which is, of course, a subset of every The Cartesian product, X $\times$ Y, is also a relation. It is obviously the largest relation from \textbf{The Definition of Binary Relation} Given a set of objects , a binary relation is a subset of the Cartesian Product . A \textbf{binary relation} from a set \textbf{\textit{A}} to a set \textbf{\textit{B}} is a set of \textbf{} \textbf{} \textbf{} \textbf{} \textbf{} \textbf{} \textbf{Definition of Cartesian product} \textbf{} The set of all ordered pairs \textbf{$<$\textit{a, b}$>$}, where \textbf{\textit{a}} is an element \textbf{The Definition of Equivalence Relation} An Equivalence Relation is a binary relation between two elements of a set which groups them together An equivalence relation is reflexive, symmetric, and transitive. In other words, for all elements Reflexivity: a \~{} a Symmetry: if a \~{} b then b \~{} a Transitivity: if a \~{} b and b \~{} c then a \~{} c. Reflexivity: A binary relation can have, among other properties, reflexivity or irreflexivity. At least in this context, (binary) relation (on X) always means a relation on X$\times$X, or in oth A reflexive relation R on set X is one where for all a in X, a is R-related to itself. In mathematical . . The reflexive closure R = is defined as R = = \{(x, x) \textbar x ? X\} ? R, i.e., the smallest re Symmetry: a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related In mathematical notation, this is: Transitivity: A binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is To write this in predicate logic: For instance, the "greater than" relation is transitive: If A $>$ B, and B $>$ C, then A $>$ C. For example, "is greater than," "is at least as great as," and "is equal to" ( equality) are transitive whenever A $>$ B and B $>$ C, then also A $>$ C whenever A = B and B = C, then also A = C whenever A = B and B = C, then also A = C \textbf{} \textbf{Partition: Definition:} A partition of a set X is a division of X into non-overlapping "parts" or "blocks" or "cells" that A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly Equivalently, a set P of subsets of X, is a partition of X if No element of P is empty. The union of the elements of P is equal to X. The intersection of any two elements of P is empty. The elements of P are sometimes called the blocks or parts of the partition. Every singleton set \{x\} has exactly one partition, namely \{ \{x\} \}. For any nonempty set X, P = \{X\} is a partition of X. The empty set has exactly one partition, namely one with no blocks. For any non-empty proper subset A of a set U, this A together with its complement is a partition of If we do not use axiom 1, then the above example generalizes so that any subset (empty or not) toge The set \{ 1, 2, 3 \} has these five partitions. \{ \{1\}, \{2\}, \{3\} \}, sometimes denoted by 1/2/3. \{ \{1, 2\}, \{3\} \}, sometimes denoted by 12/3. \{ \{1, 3\}, \{2\} \}, sometimes denoted by 13/2. \{ \{1\}, \{2, 3\} \}, sometimes denoted by 1/23. \{ \{1, 2, 3\} \}, sometimes denoted by 123. Note that \{ \{\}, \{1,3\}, \{2\} \} is not a partition if we are using axiom 1 (because it contains the empty \{ \{1,2\}, \{2, 3\} \} is not a partition (of any set) because the element 2 is contained in more \{ \{1\}, \{2\} \} is not a partition of \{1, 2, 3\} because none of its blocks contains 3; however [edit ] Partitions and equivalence relations If an equivalence relation is given on the set X, then the set of all equivalence classes forms a \textbf{} \end{document}