The document discusses relations and their representations. It defines a binary relation as a subset of A×B where A and B are nonempty sets. Relations can be represented using arrow diagrams, directed graphs, and zero-one matrices. A directed graph represents the elements of A as vertices and draws an edge from vertex a to b if aRb. The zero-one matrix representation assigns 1 to the entry in row a and column b if (a,b) is in the relation, and 0 otherwise. The document also discusses indegrees, outdegrees, composite relations, and properties of relations like reflexivity.
The document discusses closures of relations, including reflexive closure and symmetric closure. It provides definitions and theorems related to closures. It also uses an example to illustrate finding the reflexive closure and symmetric closure of a relation. Additionally, it covers topics like paths in directed graphs, shortest paths, and transitive closure. It includes an example of calculating the transitive closure of a relation by finding its zero-one matrix.
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
This document contains lecture notes on relations from a Discrete Structures course. It defines what a relation is and provides examples of relations on sets. It then discusses various properties of relations such as reflexive, symmetric, antisymmetric, transitive, and how to combine relations using set operations. It also introduces the concept of the closure of a relation and provides examples of finding the reflexive and transitive closure of relations. Finally, it provides a brief definition of what a graph is in terms of vertices and edges.
Relations and their Properties
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 9, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses relations and how they can be used to represent relationships between elements of different sets. It defines binary relations as subsets of the Cartesian product of two sets A and B, and uses notation like aRb to indicate that the ordered pair (a,b) is in the relation R. Examples of binary relations include functions and relations on a single set. The document also discusses n-ary relations, and how they can be used to represent databases as relations between tuples of fields. Key concepts for databases represented as relations include primary keys, composite keys, and relational operations like projection and join.
The document discusses different types of relations between elements of sets. It defines relations as subsets of Cartesian products of sets and describes how relations can be represented using matrices or directed graphs. It then introduces various properties of relations such as reflexive, symmetric, transitive, and defines what it means for a relation to have each property. Composition of relations is also covered, along with how relation composition can be represented by matrix multiplication.
This document discusses binary relations and their properties. A binary relation R from sets A to B is a subset of the Cartesian product A × B. Relations can be represented using matrices, where the (i,j) entry is 1 if the element (i,j) is in the relation and 0 otherwise. The properties of relations discussed include: reflexive (an element is related to itself), symmetric (if a is related to b then b is related to a), antisymmetric (if a is related to b and b is related to a then a=b), and transitive (if a is related to b and b is related to c then a is related to c). Boolean operations can be used on the matrices
The document discusses relations and their properties. It begins by defining a relation as a subset of the Cartesian product of two sets. Relations can be represented using ordered pairs in a set or graphically using arrows. Properties of relations such as reflexive, symmetric, and transitive are introduced. Examples are provided to illustrate relations and calculating their properties. The document also discusses n-ary relations, representing relations using matrices, and operations on relations such as selection.
The document discusses closures of relations, including reflexive closure and symmetric closure. It provides definitions and theorems related to closures. It also uses an example to illustrate finding the reflexive closure and symmetric closure of a relation. Additionally, it covers topics like paths in directed graphs, shortest paths, and transitive closure. It includes an example of calculating the transitive closure of a relation by finding its zero-one matrix.
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
This document contains lecture notes on relations from a Discrete Structures course. It defines what a relation is and provides examples of relations on sets. It then discusses various properties of relations such as reflexive, symmetric, antisymmetric, transitive, and how to combine relations using set operations. It also introduces the concept of the closure of a relation and provides examples of finding the reflexive and transitive closure of relations. Finally, it provides a brief definition of what a graph is in terms of vertices and edges.
Relations and their Properties
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 9, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses relations and how they can be used to represent relationships between elements of different sets. It defines binary relations as subsets of the Cartesian product of two sets A and B, and uses notation like aRb to indicate that the ordered pair (a,b) is in the relation R. Examples of binary relations include functions and relations on a single set. The document also discusses n-ary relations, and how they can be used to represent databases as relations between tuples of fields. Key concepts for databases represented as relations include primary keys, composite keys, and relational operations like projection and join.
The document discusses different types of relations between elements of sets. It defines relations as subsets of Cartesian products of sets and describes how relations can be represented using matrices or directed graphs. It then introduces various properties of relations such as reflexive, symmetric, transitive, and defines what it means for a relation to have each property. Composition of relations is also covered, along with how relation composition can be represented by matrix multiplication.
This document discusses binary relations and their properties. A binary relation R from sets A to B is a subset of the Cartesian product A × B. Relations can be represented using matrices, where the (i,j) entry is 1 if the element (i,j) is in the relation and 0 otherwise. The properties of relations discussed include: reflexive (an element is related to itself), symmetric (if a is related to b then b is related to a), antisymmetric (if a is related to b and b is related to a then a=b), and transitive (if a is related to b and b is related to c then a is related to c). Boolean operations can be used on the matrices
The document discusses relations and their properties. It begins by defining a relation as a subset of the Cartesian product of two sets. Relations can be represented using ordered pairs in a set or graphically using arrows. Properties of relations such as reflexive, symmetric, and transitive are introduced. Examples are provided to illustrate relations and calculating their properties. The document also discusses n-ary relations, representing relations using matrices, and operations on relations such as selection.
Representing Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 14, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Discrete Mathematics - Relations. ... Relations may exist between objects of the same set or between objects of two or more sets. Definition and Properties. A binary relation R from set x to y (written as x R y o r R ( x , y ) ) is a subset of the Cartesian product x × y .
The document defines an equivalence relation as a binary relation R on a set A that is reflexive, symmetric, and transitive. It provides examples to illustrate each property: reflexive means each element is related to itself, symmetric means if a is related to b then b is related to a, and transitive means if a is related to b and b is related to c then a is related to c. The document concludes with an example set and relation that demonstrates all three properties and is therefore an equivalence relation.
The document discusses relations and some of their properties. A relation R on a set A relates elements of A to each other or to elements of another set. Relations can be reflexive, symmetric, antisymmetric, transitive, etc. based on how the elements are related. The number of possible relations on a set A with n elements is 2^n^2. Relations can be combined using set operations like union and intersection. The composite of two relations R and S relates elements where there is an element in both R and S.
A relation maps elements from one set to another set through ordered pairs. The domain is the set of first elements in the ordered pairs and the range is the set of second elements. Relations can have properties like being reflexive, symmetric, transitive, or an equivalence relation. Relations are used in applications like relational databases, project scheduling, and communication networks.
Relations represent relationships between elements of sets. Binary relations relate elements of two sets A and B, and are represented as subsets of the Cartesian product A × B. N-ary relations relate elements of more than two sets. Relations can be represented using set notation, arrow diagrams, matrices, or coordinate systems. A relation is reflexive if each element is related to itself, symmetric if aRb implies bRa, and transitive if aRb and bRc imply aRc. Relations can be combined using set operations like union and intersection. The composite of relations R and S relates a to c if there exists a b such that a is related to b by R and b is related to c by S.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
BCA_Semester-II-Discrete Mathematics_unit-ii_Relation and orderingRai University
This document defines and explains various concepts related to relations and ordering in discrete mathematics including:
- A relation is a set of ordered pairs where the first item is the domain and second is the range.
- Relations can be binary, reflexive, symmetric, transitive, equivalence relations and partial orders.
- Equivalence classes are sets of equivalent elements under an equivalence relation.
- Graphs and matrices can represent relations. Hasse diagrams show partial orderings visually.
- Upper and lower bounds, maximal/minimal elements, chains and covers are discussed in the context of partial orders.
This document introduces the concept of relations in mathematics. It defines a relation as a subset of the Cartesian product between two sets that describes a connection between the ordered pairs. It discusses the domain and range of a relation, and how relations can be represented algebraically or through arrow diagrams. Examples are given of different types of relations such as reflexive, symmetric, and transitive relations. Equivalence relations and equivalence classes are also introduced.
relasi dan fungsi kelompok 4 smpn 3 cikarang timur kelas 83Dewi Dewi
1. The document is a presentation by Group 4 from SMPN 3 Cikarang Timur about relations and functions.
2. It defines relations and functions, and ways to represent them using arrow diagrams, Cartesian diagrams, and ordered pairs.
3. Examples are given of determining the domain, codomain, and range of functions from diagrams or equations. Formulas for functions are also derived.
This document discusses relations and cartesian products of sets. It defines a relation as a subset of the cartesian product of two sets that links elements between the sets. It provides examples of forming cartesian products of different sets to generate ordered pairs, and defines relations between the elements of those pairs based on given conditions. The key concepts covered are cartesian products, ordered pairs, domains and ranges of relations, and representing relations using arrow diagrams or set-builder and roster forms.
This document discusses relation matrices and graphs. It begins by defining a relation matrix as a way to represent a relation between two finite sets A and B using a matrix with 1s and 0s. An example is provided to demonstrate how to construct a relation matrix. The document then discusses how relations can be represented using graphs by connecting elements with edges. Properties of relations like reflexive, symmetric, and anti-symmetric are explained through examples using relation matrices. Finally, the conclusion restates that relation matrices and graphs can be used to represent relations between sets.
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 1/ Theory of Cartesian product Relation in a Set Types of Relations Equivalence Relation and Equivalence Class explained with examples
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 2/ Types of relations/ Reflexive Symmetric and Transitive relations/ Equivalence relation/ Equivalence class
Stability criterion of periodic oscillations in a (9)Alexander Decker
This document studies the fixed points and properties of the Duffing map, which is a 2-D discrete dynamical system. It finds that the Duffing map has three fixed points when a > b + 1. It divides the parameter space into six regions to determine whether the fixed points are attracting, repelling, or saddle points. It also determines the bifurcation points of the parameter space. Key properties of the Duffing map explored include that it is a diffeomorphism when b ≠ 0, its Jacobian is equal to b, and the eigenvalues of its derivative depend on a and b.
The document discusses partial ordered sets (POSETs). It begins by defining a POSET as a set A together with a partial order R, which is a relation on A that is reflexive, antisymmetric, and transitive. An example is given of the set of integers under the relation "greater than or equal to". It is shown that this relation satisfies the three properties of a partial order. The document emphasizes that a relation must satisfy all three properties - reflexive, antisymmetric, and transitive - to be considered a partial order. Some example relations on a set are provided and it is discussed which of these are partial orders.
This document contains information from a class presentation on computing elements of one-dimensional arrays and recursively defined sequences. It discusses how to count the elements of an array that is cut in the middle, the probability of an element having an even or odd subscript, computing terms of a recursively defined sequence, and properties of relations. The presentation was given by 5 students to their instructor for their BS in Computer Science class from 2013-2017.
Brahmagupta was a distinguished 7th century mathematician and astronomer from India. He lived and worked in Ujjain, where he wrote his first book Brahmasphuta-siddhanta at the age of 30. The book contains knowledge on arithmetic, geometry, algebra and astronomy. Some of Brahmagupta's key contributions include being the first to use negative numbers and define zero, giving methods for multiplication, solving quadratic equations, and formulas for finding properties of triangles and cyclic quadrilaterals. He was an influential mathematician who applied algebra to astronomy.
This document discusses mathematical concepts related to relations including:
1. The inverse of a relation R-1, which relates elements in the opposite direction as R.
2. The composition of two relations R and S, denoted R◦S or RS, which relates elements related by both R and S.
3. Matrices can represent relations and be used to calculate their composition.
4. A partial order relation on a set A is a relation that is reflexive, antisymmetric, and transitive. Examples of partial order relations include set inclusion and the less than or equal to relation on real numbers.
This document defines and provides examples of binary relations and properties of relations. A binary relation R from set A to set B is a subset of the Cartesian product A x B that contains ordered pairs (a,b) where a is from A and b is from B. Examples demonstrate relations between people and cars they drive as well as subsets of Cartesian products that do or do not qualify as binary relations. The document also defines properties of relations such as symmetric, antisymmetric, asymmetric, transitive, and reflexive, and provides examples of relations that do or do not satisfy each property.
Representing Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 14, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Discrete Mathematics - Relations. ... Relations may exist between objects of the same set or between objects of two or more sets. Definition and Properties. A binary relation R from set x to y (written as x R y o r R ( x , y ) ) is a subset of the Cartesian product x × y .
The document defines an equivalence relation as a binary relation R on a set A that is reflexive, symmetric, and transitive. It provides examples to illustrate each property: reflexive means each element is related to itself, symmetric means if a is related to b then b is related to a, and transitive means if a is related to b and b is related to c then a is related to c. The document concludes with an example set and relation that demonstrates all three properties and is therefore an equivalence relation.
The document discusses relations and some of their properties. A relation R on a set A relates elements of A to each other or to elements of another set. Relations can be reflexive, symmetric, antisymmetric, transitive, etc. based on how the elements are related. The number of possible relations on a set A with n elements is 2^n^2. Relations can be combined using set operations like union and intersection. The composite of two relations R and S relates elements where there is an element in both R and S.
A relation maps elements from one set to another set through ordered pairs. The domain is the set of first elements in the ordered pairs and the range is the set of second elements. Relations can have properties like being reflexive, symmetric, transitive, or an equivalence relation. Relations are used in applications like relational databases, project scheduling, and communication networks.
Relations represent relationships between elements of sets. Binary relations relate elements of two sets A and B, and are represented as subsets of the Cartesian product A × B. N-ary relations relate elements of more than two sets. Relations can be represented using set notation, arrow diagrams, matrices, or coordinate systems. A relation is reflexive if each element is related to itself, symmetric if aRb implies bRa, and transitive if aRb and bRc imply aRc. Relations can be combined using set operations like union and intersection. The composite of relations R and S relates a to c if there exists a b such that a is related to b by R and b is related to c by S.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
BCA_Semester-II-Discrete Mathematics_unit-ii_Relation and orderingRai University
This document defines and explains various concepts related to relations and ordering in discrete mathematics including:
- A relation is a set of ordered pairs where the first item is the domain and second is the range.
- Relations can be binary, reflexive, symmetric, transitive, equivalence relations and partial orders.
- Equivalence classes are sets of equivalent elements under an equivalence relation.
- Graphs and matrices can represent relations. Hasse diagrams show partial orderings visually.
- Upper and lower bounds, maximal/minimal elements, chains and covers are discussed in the context of partial orders.
This document introduces the concept of relations in mathematics. It defines a relation as a subset of the Cartesian product between two sets that describes a connection between the ordered pairs. It discusses the domain and range of a relation, and how relations can be represented algebraically or through arrow diagrams. Examples are given of different types of relations such as reflexive, symmetric, and transitive relations. Equivalence relations and equivalence classes are also introduced.
relasi dan fungsi kelompok 4 smpn 3 cikarang timur kelas 83Dewi Dewi
1. The document is a presentation by Group 4 from SMPN 3 Cikarang Timur about relations and functions.
2. It defines relations and functions, and ways to represent them using arrow diagrams, Cartesian diagrams, and ordered pairs.
3. Examples are given of determining the domain, codomain, and range of functions from diagrams or equations. Formulas for functions are also derived.
This document discusses relations and cartesian products of sets. It defines a relation as a subset of the cartesian product of two sets that links elements between the sets. It provides examples of forming cartesian products of different sets to generate ordered pairs, and defines relations between the elements of those pairs based on given conditions. The key concepts covered are cartesian products, ordered pairs, domains and ranges of relations, and representing relations using arrow diagrams or set-builder and roster forms.
This document discusses relation matrices and graphs. It begins by defining a relation matrix as a way to represent a relation between two finite sets A and B using a matrix with 1s and 0s. An example is provided to demonstrate how to construct a relation matrix. The document then discusses how relations can be represented using graphs by connecting elements with edges. Properties of relations like reflexive, symmetric, and anti-symmetric are explained through examples using relation matrices. Finally, the conclusion restates that relation matrices and graphs can be used to represent relations between sets.
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 1/ Theory of Cartesian product Relation in a Set Types of Relations Equivalence Relation and Equivalence Class explained with examples
JEE Mathematics/ Lakshmikanta Satapathy/ Relations and Functions theory part 2/ Types of relations/ Reflexive Symmetric and Transitive relations/ Equivalence relation/ Equivalence class
Stability criterion of periodic oscillations in a (9)Alexander Decker
This document studies the fixed points and properties of the Duffing map, which is a 2-D discrete dynamical system. It finds that the Duffing map has three fixed points when a > b + 1. It divides the parameter space into six regions to determine whether the fixed points are attracting, repelling, or saddle points. It also determines the bifurcation points of the parameter space. Key properties of the Duffing map explored include that it is a diffeomorphism when b ≠ 0, its Jacobian is equal to b, and the eigenvalues of its derivative depend on a and b.
The document discusses partial ordered sets (POSETs). It begins by defining a POSET as a set A together with a partial order R, which is a relation on A that is reflexive, antisymmetric, and transitive. An example is given of the set of integers under the relation "greater than or equal to". It is shown that this relation satisfies the three properties of a partial order. The document emphasizes that a relation must satisfy all three properties - reflexive, antisymmetric, and transitive - to be considered a partial order. Some example relations on a set are provided and it is discussed which of these are partial orders.
This document contains information from a class presentation on computing elements of one-dimensional arrays and recursively defined sequences. It discusses how to count the elements of an array that is cut in the middle, the probability of an element having an even or odd subscript, computing terms of a recursively defined sequence, and properties of relations. The presentation was given by 5 students to their instructor for their BS in Computer Science class from 2013-2017.
Brahmagupta was a distinguished 7th century mathematician and astronomer from India. He lived and worked in Ujjain, where he wrote his first book Brahmasphuta-siddhanta at the age of 30. The book contains knowledge on arithmetic, geometry, algebra and astronomy. Some of Brahmagupta's key contributions include being the first to use negative numbers and define zero, giving methods for multiplication, solving quadratic equations, and formulas for finding properties of triangles and cyclic quadrilaterals. He was an influential mathematician who applied algebra to astronomy.
This document discusses mathematical concepts related to relations including:
1. The inverse of a relation R-1, which relates elements in the opposite direction as R.
2. The composition of two relations R and S, denoted R◦S or RS, which relates elements related by both R and S.
3. Matrices can represent relations and be used to calculate their composition.
4. A partial order relation on a set A is a relation that is reflexive, antisymmetric, and transitive. Examples of partial order relations include set inclusion and the less than or equal to relation on real numbers.
This document defines and provides examples of binary relations and properties of relations. A binary relation R from set A to set B is a subset of the Cartesian product A x B that contains ordered pairs (a,b) where a is from A and b is from B. Examples demonstrate relations between people and cars they drive as well as subsets of Cartesian products that do or do not qualify as binary relations. The document also defines properties of relations such as symmetric, antisymmetric, asymmetric, transitive, and reflexive, and provides examples of relations that do or do not satisfy each property.
The document defines and provides examples of relational algebra concepts. It explains that a relation is a subset of ordered pairs from two sets that represents a connection between the pairs' elements. A binary relation from set A to set B is a subset of the Cartesian product of A and B. The inverse of a relation R consists of reversing the ordered pairs in R. Relations can be represented visually using matrices or directed graphs.
This document provides an overview of chapter 2 on relations from a discrete mathematics course. It defines key concepts such as product sets, relations, inverse relations, representing relations using matrices, and composition of relations. It also covers different types of relations like reflexive, symmetric, antisymmetric, transitive, and equivalence relations. Examples are provided to illustrate these concepts and determine if specific relations satisfy the given properties. The document is copyrighted material from a course on discrete mathematics taught in 2014-2015.
A relation is a set of ordered pairs that shows a relationship between elements of two sets. An ordered pair connects an element from one set to an element of another set. The domain of a relation is the set of first elements of each ordered pair, while the range is the set of second elements. Relations can be represented visually using arrow diagrams or directed graphs to show the connections between elements of different sets defined by the relation.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
This document discusses relations and various types of relations. It begins by defining what a relation is as a subset of the Cartesian product of two sets and provides examples of relations. It then discusses the domain and range of relations and inverse relations. The document outlines several types of relations including reflexive, irreflexive, symmetric, and transitive relations and provides examples of each. It concludes by discussing the objectives of understanding different types of relations and their properties.
the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown the quick brown
This document provides an overview of relations and their properties in discrete mathematics. It defines what a relation is, distinguishes between relations and functions, and describes key properties of relations including:
- Reflexive relations, where every element is related to itself.
- Symmetric relations, where if a is related to b then b is related to a.
- Transitive relations, where if a is related to b and b is related to c, then a is related to c.
It also discusses how to determine if a relation has these properties, combines multiple relations using set operations, and defines the composite of two relations. The overall goal is for students to understand relations and be able to analyze them for
The document discusses relations and functions. It defines relations as subsets of Cartesian products of sets and describes how to classify relations as reflexive, symmetric, transitive, or an equivalence relation. It also defines functions, including their domain, codomain, and range. It describes how to classify functions as injective, surjective, or bijective. Examples are provided to illustrate these concepts of relations and functions.
(1) The document discusses relations and functions in mathematics. It defines different types of relations such as empty relation, universal relation, equivalence relation, reflexive relation, symmetric relation and transitive relation. (2) It provides examples to illustrate these relations and checks whether given relations satisfy the properties. (3) The document also discusses that an equivalence relation partitions a set into mutually exclusive equivalence classes.
The document discusses binary relations and properties of relations. A binary relation R from set A to set B is a subset of the Cartesian product A × B. Relations can model real-world relationships between elements, like which people drive which cars. Functions are a special type of binary relation where each element of the domain A is related to exactly one element of the range B. The properties of relations discussed include reflexive, symmetric, antisymmetric, transitive, and combining relations using operations like union, intersection, and composition.
This document provides definitions and examples of relations and different types of relations. It discusses relations as sets of ordered pairs that satisfy a given rule or property. Reflexive, symmetric, and transitive relations are defined. Several examples of relations over different sets are given and determined to be reflexive, symmetric, transitive or none of the above. Solutions to exercises involving checking properties of various relations are also provided.
This document discusses set theory and relations between sets. It begins by introducing basic set notation such as set membership and subset notation. It then defines and provides examples of relations between sets such as subset, equality, union, intersection, difference, and complement. The document also covers properties of sets and relations including commutative, associative, distributive, and other properties. It concludes by discussing relations as subsets of Cartesian products and properties of relations such as reflexive, symmetric, transitive, and antisymmetric relations.
This document defines and explains various concepts related to sets and relations. It discusses the four main set operations of union, intersection, complement, and difference. It then explains eight types of relations: empty, universal, identity, inverse, reflexive, symmetric, transitive, and equivalence relations. Finally, it defines partial ordering as a relation that is reflexive, antisymmetric, and transitive.
This document is a lecture on relations and functions from a class on Applied Mathematics 1. It introduces key concepts like Cartesian products, relations, inverse relations, different ways to represent relations visually, and composition of relations. Specifically, it defines relations as subsets of Cartesian products, discusses inverse relations as reversing ordered pairs, and defines the composition of two relations R and S as the set of all ordered pairs (a,c) where there exists a b such that (a,b) is in R and (b,c) is in S. Several examples are provided to illustrate these concepts.
This document defines and provides examples of relations and properties of relations. It begins by defining the cartesian product of two sets A and B as the set of all ordered pairs (x,y) where x is in A and y is in B. It notes that the order of sets matters, so A×B is not equal to B×A. It then defines a relation R from set A to B as a subset of the cartesian product A×B.
The document goes on to define properties of relations - reflexive, symmetric, and transitive - and provides examples of relations that satisfy each of these properties. It notes that a relation is an equivalence relation if it is reflexive, symmetric, and transitive.
The document discusses types of relations. It defines empty relation, universal relation, reflexive relation, symmetric relation and transitive relation. An equivalence relation is a relation that is reflexive, symmetric and transitive. An equivalence relation partitions a set into mutually exclusive and exhaustive equivalence classes. Elements within an equivalence class are related to each other under the relation, while elements of different classes are not related. Examples of equivalence relations and their equivalence classes are provided.
A set is an unordered collection of unique elements. A set can be written explicitly using set brackets. The order of elements in a set does not matter. Some key concepts about sets discussed in the document include: types of sets like finite, empty, singleton, infinite sets; cardinality which is the number of elements in a set; set operations like intersection, union, complement; and set relations which describe connections between elements of different sets using ordered pairs. Common relations include empty, full, identity, inverse, symmetric, transitive, and equivalence relations.
The document provides an introduction to unsupervised learning and reinforcement learning. It then discusses eigen values and eigen vectors, showing how to calculate them from a matrix. It provides examples of covariance matrices and using Gaussian elimination to solve for eigen vectors. Finally, it discusses principal component analysis and different clustering algorithms like K-means clustering.
Cross validation is a technique for evaluating machine learning models by splitting the dataset into training and validation sets and training the model multiple times on different splits, to reduce variance. K-fold cross validation splits the data into k equally sized folds, where each fold is used once for validation while the remaining k-1 folds are used for training. Leave-one-out cross validation uses a single observation from the dataset as the validation set. Stratified k-fold cross validation ensures each fold has the same class proportions as the full dataset. Grid search evaluates all combinations of hyperparameters specified as a grid, while randomized search samples hyperparameters randomly within specified ranges. Learning curves show training and validation performance as a function of training set size and can diagnose underfitting
This document provides an overview of supervised machine learning algorithms for classification, including logistic regression, k-nearest neighbors (KNN), support vector machines (SVM), and decision trees. It discusses key concepts like evaluation metrics, performance measures, and use cases. For logistic regression, it covers the mathematics behind maximum likelihood estimation and gradient descent. For KNN, it explains the algorithm and discusses distance metrics and a numerical example. For SVM, it outlines the concept of finding the optimal hyperplane that maximizes the margin between classes.
The document provides information on solving the sum of subsets problem using backtracking. It discusses two formulations - one where solutions are represented by tuples indicating which numbers are included, and another where each position indicates if the corresponding number is included or not. It shows the state space tree that represents all possible solutions for each formulation. The tree is traversed depth-first to find all solutions where the sum of the included numbers equals the target sum. Pruning techniques are used to avoid exploring non-promising paths.
The document discusses the greedy method and its applications. It begins by defining the greedy approach for optimization problems, noting that greedy algorithms make locally optimal choices at each step in hopes of finding a global optimum. Some applications of the greedy method include the knapsack problem, minimum spanning trees using Kruskal's and Prim's algorithms, job sequencing with deadlines, and finding the shortest path using Dijkstra's algorithm. The document then focuses on explaining the fractional knapsack problem and providing a step-by-step example of solving it using a greedy approach. It also provides examples and explanations of Kruskal's algorithm for finding minimum spanning trees.
The document describes various divide and conquer algorithms including binary search, merge sort, quicksort, and finding maximum and minimum elements. It begins by explaining the general divide and conquer approach of dividing a problem into smaller subproblems, solving the subproblems independently, and combining the solutions. Several examples are then provided with pseudocode and analysis of their divide and conquer implementations. Key algorithms covered in the document include binary search (log n time), merge sort (n log n time), and quicksort (n log n time on average).
What is an Algorithm
Time Complexity
Space Complexity
Asymptotic Notations
Recursive Analysis
Selection Sort
Insertion Sort
Recurrences
Substitution Method
Master Tree Method
Recursion Tree Method
This document provides an outline for a machine learning syllabus. It includes 14 modules covering topics like machine learning terminology, supervised and unsupervised learning algorithms, optimization techniques, and projects. It lists software and hardware requirements for the course. It also discusses machine learning applications, issues, and the steps to build a machine learning model.
The document discusses problem-solving agents and their approach to solving problems. Problem-solving agents (1) formulate a goal based on the current situation, (2) formulate the problem by defining relevant states and actions, and (3) search for a solution by exploring sequences of actions that lead to the goal state. Several examples of problems are provided, including the 8-puzzle, robotic assembly, the 8 queens problem, and the missionaries and cannibals problem. For each problem, the relevant states, actions, goal tests, and path costs are defined.
The simplex method is a linear programming algorithm that can solve problems with more than two decision variables. It works by generating a series of solutions, called tableaus, where each tableau corresponds to a corner point of the feasible solution space. The algorithm starts at the initial tableau, which corresponds to the origin. It then shifts to adjacent corner points, moving in the direction that optimizes the objective function. This process of generating new tableaus continues until an optimal solution is found.
The document discusses functions and the pigeonhole principle. It defines what a function is, how functions can be represented graphically and with tables and ordered pairs. It covers one-to-one, onto, and bijective functions. It also discusses function composition, inverse functions, and the identity function. The pigeonhole principle states that if n objects are put into m containers where n > m, then at least one container must hold more than one object. Examples are given to illustrate how to apply the principle to problems involving months, socks, and selecting numbers.
This document discusses logic and propositional logic. It covers the following topics:
- The history and applications of logic.
- Different types of statements and their grammar.
- Propositional logic including symbols, connectives, truth tables, and semantics.
- Quantifiers, universal and existential quantification, and properties of quantifiers.
- Normal forms such as disjunctive normal form and conjunctive normal form.
- Inference rules and the principle of mathematical induction, illustrated with examples.
1. Set theory is an important mathematical concept and tool that is used in many areas including programming, real-world applications, and computer science problems.
2. The document introduces some basic concepts of set theory including sets, members, operations on sets like union and intersection, and relationships between sets like subsets and complements.
3. Infinite sets are discussed as well as different types of infinite sets including countably infinite and uncountably infinite sets. Special sets like the empty set and power sets are also covered.
The document discusses uncertainty and probabilistic reasoning. It describes sources of uncertainty like partial information, unreliable information, and conflicting information from multiple sources. It then discusses representing and reasoning with uncertainty using techniques like default logic, rules with probabilities, and probability theory. The key approaches covered are conditional probability, independence, conditional independence, and using Bayes' rule to update probabilities based on new evidence.
The document outlines the objectives, outcomes, and learning outcomes of a course on artificial intelligence. The objectives include conceptualizing ideas and techniques for intelligent systems, understanding mechanisms of intelligent thought and action, and understanding advanced representation and search techniques. Outcomes include developing an understanding of AI building blocks, choosing appropriate problem solving methods, analyzing strengths and weaknesses of AI approaches, and designing models for reasoning with uncertainty. Learning outcomes include knowledge, intellectual skills, practical skills, and transferable skills in artificial intelligence.
Planning involves representing an initial state, possible actions, and a goal state. A planning agent uses a knowledge base to select action sequences that transform the initial state into a goal state. STRIPS is a common planning representation that uses predicates to describe states and logical operators to represent actions and their effects. A STRIPS planning problem specifies the initial state, goal conditions, and set of operators. A solution is a sequence of ground operator instances that produces the goal state from the initial state.
Cricket management system ptoject report.pdfKamal Acharya
The aim of this project is to provide the complete information of the National and
International statistics. The information is available country wise and player wise. By
entering the data of eachmatch, we can get all type of reports instantly, which will be
useful to call back history of each player. Also the team performance in each match can
be obtained. We can get a report on number of matches, wins and lost.
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation w...IJCNCJournal
Paper Title
Particle Swarm Optimization–Long Short-Term Memory based Channel Estimation with Hybrid Beam Forming Power Transfer in WSN-IoT Applications
Authors
Reginald Jude Sixtus J and Tamilarasi Muthu, Puducherry Technological University, India
Abstract
Non-Orthogonal Multiple Access (NOMA) helps to overcome various difficulties in future technology wireless communications. NOMA, when utilized with millimeter wave multiple-input multiple-output (MIMO) systems, channel estimation becomes extremely difficult. For reaping the benefits of the NOMA and mm-Wave combination, effective channel estimation is required. In this paper, we propose an enhanced particle swarm optimization based long short-term memory estimator network (PSOLSTMEstNet), which is a neural network model that can be employed to forecast the bandwidth required in the mm-Wave MIMO network. The prime advantage of the LSTM is that it has the capability of dynamically adapting to the functioning pattern of fluctuating channel state. The LSTM stage with adaptive coding and modulation enhances the BER.PSO algorithm is employed to optimize input weights of LSTM network. The modified algorithm splits the power by channel condition of every single user. Participants will be first sorted into distinct groups depending upon respective channel conditions, using a hybrid beamforming approach. The network characteristics are fine-estimated using PSO-LSTMEstNet after a rough approximation of channels parameters derived from the received data.
Keywords
Signal to Noise Ratio (SNR), Bit Error Rate (BER), mm-Wave, MIMO, NOMA, deep learning, optimization.
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Sachpazis_Consolidation Settlement Calculation Program-The Python Code and th...Dr.Costas Sachpazis
Consolidation Settlement Calculation Program-The Python Code
By Professor Dr. Costas Sachpazis, Civil Engineer & Geologist
This program calculates the consolidation settlement for a foundation based on soil layer properties and foundation data. It allows users to input multiple soil layers and foundation characteristics to determine the total settlement.
This is an overview of my current metallic design and engineering knowledge base built up over my professional career and two MSc degrees : - MSc in Advanced Manufacturing Technology University of Portsmouth graduated 1st May 1998, and MSc in Aircraft Engineering Cranfield University graduated 8th June 2007.
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Data Communication and Computer Networks Management System Project Report.pdfKamal Acharya
Networking is a telecommunications network that allows computers to exchange data. In
computer networks, networked computing devices pass data to each other along data
connections. Data is transferred in the form of packets. The connections between nodes are
established using either cable media or wireless media.
2. Relations
If we want to describe a relationship between elements of two
sets A and B, we can use ordered pairs with their first
element taken from A and their second element taken from B.
Since this is a relation between two sets, it is called a binary
relation.
Definition: Let A and B be nonempty sets. A binary relation
from A to B is a subset of AB.
In other words, for a binary relation R we have
R AB. We use the notation aRb to denote that (a, b)R
and to denote that (a, b)R.
Note: If A=B, we say that R ⊆ AXA is a relation on AShiwani Gupta 2
3. Relations
❑ R can be described in
❑ Roster form
❑ Set-builder form
Shiwani Gupta 3
4. Representing Relations
❑Arrow Diagram
❑Write the elements of A in one column
❑Write the elements B in another column
❑Draw an arrow from an element, a, of A to an element, b, of B,
if (a ,b) R
❑Here, A = {2,3,5} and B = {7,10,12,30} and R from A into B is
defined as follows: For all a A and b B, a R b if and only if
a divides b
❑The symbol → (called an arrow) represents the relation R
Shiwani Gupta 4
6. Representing Relations
❑Directed Graph
❑Let R be a relation on a finite set A
❑Describe R pictorially as follows:
❑For each element of A , draw a small or big dot and label
the dot by the corresponding element of A
❑Draw an arrow from a dot labeled a , to another dot
labeled, b , if a R b .
❑Resulting pictorial representation of R is called the
directed graph representation of the relation R
Shiwani Gupta 6
8. Representing Relations
❑Directed graph (Digraph) representation of R
❑Each dot is called a vertex
❑If a vertex is labeled, a, then it is also called vertex a
❑ An arc from a vertex labeled a, to another vertex, b is
called a directed edge, or directed arc from a to b
❑The ordered pair (A , R) a directed graph, or digraph,
of the relation R, where each element of A is a called
a vertex of the digraph
Shiwani Gupta 8
9. Representing Relations
❑Directed graph (Digraph) representation of R
(Continued)
❑For vertices a and b , if a R b, a is adjacent to b and b is
adjacent from a
❑Because (a, a) R, an arc from a to a is drawn; because
(a, b) R, an arc is drawn from a to b. Similarly, arcs are
drawn from b to b, b to c , b to a, b to d, and c to d
❑For an element a A such that (a, a) R, a directed edge
is drawn from a to a. Such a directed edge is called a loop
at vertex a
Shiwani Gupta 9
10. Representing Relations
❑Directed graph (Digraph) representation of R
(Continued)
❑Position of each vertex is not important
❑In the digraph of a relation R, there is a directed edge
or arc from a vertex a to a vertex b if and only if a R b
❑Let A ={a ,b ,c ,d} and let R be the relation defined by
the following set:
R = {(a ,a ), (a ,b ), (b ,b ), (b ,c ), (b ,a ), (b ,d ), (c ,d
)}
Shiwani Gupta 10
11. Paths in relations and digraphs
❑A path of length n in R from a to b is a finite sequence
❑Π:a, x1, x2, …, xn-1, b; beginning with a and ending
with b such that aRx1, x1Rx2, …, xn-1Rb
❑A path that begins and ends at the same vertex is cycle
❑A path of length n involves n+1 elements of A, not
necessarily distinct
Shiwani Gupta 11
13. Relations
❑ Let A = {a, b, c , d, e , f , g , h, i, j }.
Let R be a relation on A such that the
digraph of R is as shown in Figure
3.14.
❑ Then a, b, c , d, e , f , c , g is a directed
walk in R as a R b,b R c,c R d,d R e, e
R f , f R c, c R g. Similarly, a, b, c , g is
also a directed walk in R. In the walk a,
b, c , d, e , f , c , g , the internal vertices
are b, c , d, e , f , and c , which are not
distinct as c repeats.
❑ This walk is not a path. In the walk a,
b, c , g , the internal vertices are b and c
, which are distinct. Therefore, the
walk a, b, c, g is a path.
Shiwani Gupta 13
14. Representing Relations
❑Zero-one matrices
If R is a relation from A = {a1, a2, …, am} to B =
{b1, b2, …, bn}, then R can be represented by the zero-one
matrix MR = [mij] with
mij = 1, if (ai, bj)R, and
mij = 0, if (ai, bj)R.
Note that for creating this matrix we first need to list the
elements in A and B in a particular, but arbitrary order.
Shiwani Gupta 14
15. Representing Relations
Example: How can we represent the relation
R = {(2, 1), (3, 1), (3, 2)} as a zero-one matrix?
Solution: The matrix MR is given by
=
11
01
00
RM
Shiwani Gupta 15
16. Representing Relations
The Boolean operations join and meet (you remember?)
can be used to determine the matrices representing the
union and the intersection of two relations, respectively.
To obtain the join of two zero-one matrices, we apply the
Boolean “or” function to all corresponding elements in the
matrices.
To obtain the meet of two zero-one matrices, we apply the
Boolean “and” function to all corresponding elements in the
matrices.
Shiwani Gupta 16
17. Representing Relations
Example: Let the relations R and S be represented by the
matrices
==
011
111
101
SRSR MMM
=
001
110
101
SM
What are the matrices representing RS and RS?
Solution: These matrices are given by
==
000
000
101
SRSR MMM
=
010
001
101
RM
Shiwani Gupta 17
18. Representing Relations
Example: How can we represent the relation
R = {(2, 1), (3, 1), (3, 2)} as a zero-one matrix?
Solution: The matrix MR is given by
=
11
01
00
RM
Shiwani Gupta 18
19. Indegree and outdegree
❑Indegree: Let R is a relation on A and a A, then
indegree of a A is the no. of b such that bRa i.e. no.
of b|(b,a) R
❑Outdegree: Let R is a relation on A and a A, then
outdegree of a A is the no. of b such that aRb i.e.
no. of b|(a,b) R
❑Eg. A={1,2,3,4,6}=B; aRb iff b is multiple of a
Find matrix relation and relation digraph
Find indegree and outdegree of each vertex
Shiwani Gupta 19
20. Relations
❑Domain and Range of the Relation
❑Let R be a relation from a set A into a set B. Then R ⊆ A x
B. The elements of the relation R tell which element of A
is R-related to which element of B
Shiwani Gupta 20
28. Composite Relations
Definition: Let R be a relation from a set A to a set B and S
a relation from B to a set C. The composite of R and S is
the relation consisting of ordered pairs (a,c), where aA,
cC, and for which there exists an element bB such that
(a,b)R and
(b,c)S. We denote the composite of R and S by
SR.
In other words, if relation R contains a pair (a,b) and
relation S contains a pair (b,c), then SR contains a pair
(a,c).
Shiwani Gupta 28
31. Composite Relations
Example: Let D and S be relations on A = {1, 2, 3, 4}.
D = {(a, b) | b = 5 - a} “b equals (5 – a)”
S = {(a, b) | a < b} “a is smaller than b”
D = {(1, 4), (2, 3), (3, 2), (4, 1)}
S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
SD = {(2, 4), (3, 3), (3, 4), (4, 2), (4, 3),
D maps an element a to the element (5 – a), and afterwards S
maps (5 – a) to all elements larger than (5 – a), resulting in
SD = {(a,b) | b > 5 – a} or SD = {(a,b) | a + b > 5}.
(4, 4)}
Shiwani Gupta 31
32. Connectivity Relation
A relation denoted by R∞ and defined by x R∞y if there is a path of any
length from x to y in R, is called connectivity relation of R
Shiwani Gupta 32
33. Representing Relations Using Matrices
Let us now assume that the zero-one matrices
MA = [aij], MB = [bij] and MC = [cij] represent relations A, B,
and C, respectively.
Remember: For MC = MAMB we have:
cij = 1 if and only if at least one of the terms
(ain bnj) = 1 for some n; otherwise cij = 0.
In terms of the relations, this means that C contains a pair (xi,
zj) if and only if there is an element yn such that (xi, yn) is in
relation A and
(yn, zj) is in relation B.
Therefore, C = BA (composite of A and B).Shiwani Gupta 33
34. Representing Relations Using Matrices
This gives us the following rule:
MBA = MAMB
In other words, the matrix representing the composite of
relations A and B is the Boolean product of the matrices
representing A and B.
Analogously, we can find matrices representing the powers
of relations:
MRn = MR
[n] (n-th Boolean power).
Shiwani Gupta 34
35. Representing Relations Using Matrices
Example: Find the matrix representing R2, where the
matrix representing R is given by
=
001
110
010
RM
Solution: The matrix for R2 is given by
==
010
111
110
]2[
2 RR
MM
Shiwani Gupta 35
36. Properties of Relations
We will now look at some useful ways to classify relations.
Definition: A relation R on a set A is called reflexive if (a, a)R for every element
aA.
Are the following relations on {1, 2, 3, 4} reflexive?
R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 4)} No.
R = {(1, 1), (2, 2), (2, 3), (3, 3), (4, 4)}
Yes.
R = {(1, 1), (2, 2), (3, 3)}
No.
Definition: A relation on a set A is called irreflexive if (a, a)R for every element
aA.
Note: Δ ⊆ R if R is reflexive relation on A
Δ ∩ R= Ø if relation R is irreflexive
Relation R may be either reflexive or irreflexive but not both
Diagonal Relation / Equality Relation : Δ = {(a, a) for every element aA}.
aRa for every a A but for any b A
Shiwani Gupta 36
37. Representing Relations
What do we know about the matrices representing a
relation on a set (a relation from A to A) ?
They are square matrices.
What do we know about matrices representing reflexive
relations?
All the elements on the diagonal of such matrices Mref must
be 1s.
=
1
.
.
.
1
1
refM
Shiwani Gupta 37
38. Properties of Relations
Definitions:
A relation R on a set A is called symmetric if (b, a)R whenever (a,
b)R for all a, bA.
A relation R on a set A is called antisymmetric if a = b whenever (a,
b)R and (b, a)R.
A relation R on a set A is called asymmetric if (a, b)R implies that (b,
a)R for all (a, b)A.
A relation R on a set A is called not symmetric if (a, b)R but (b,
a)R for some (a, b)A.
Note: R-1=R for symmetric relation
R-1 ≠ R for not symmetric relation
R-1 ∩ R= Ø for asymmetric relation
R-1 ∩ R ⊆ Δ for anti symmetric relationShiwani Gupta 38
39. Representing Relations
What do we know about the matrices representing
symmetric relations?
These matrices are symmetric, that is, MR = (MR)t.
=
1101
1001
0010
1101
RM
symmetric matrix,
symmetric relation.
=
0011
0011
0011
0011
RM
non-symmetric matrix,
non-symmetric relation.
Shiwani Gupta 39
40. Properties of Relations
Are the following relations on {1, 2, 3, 4}
symmetric, antisymmetric, or asymmetric?
R = {(1, 1), (1, 2), (2, 1), (3, 3), (4, 4)} symmetric
R = {(1, 1)} sym. and
antisym.
R = {(1, 3), (3, 2), (2, 1)} antisym. and
asym.
R = {(4, 4), (3, 3), (1, 4)} antisym.
Shiwani Gupta 40
41. Properties of Relations
Definition: A relation R on a set A is called transitive if whenever
(a, b)R and (b, c)R, then (a, c)R for a, b, cA.
Are the following relations on {1, 2, 3, 4} transitive?
R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)} Yes.
R = {(1, 3), (3, 2), (2, 1)} No.
R = {(2, 4), (4, 3), (2, 3), (4, 1)} No.
Shiwani Gupta 41
53. Equivalence Classes
Definition: Let R be an equivalence relation on a set A.
The set of all elements that are related to an element a of A
is called the equivalence class
of a.
The equivalence class of a with respect to R is denoted by
[a]R.
When only one relation is under consideration, we will
delete the subscript R and write [a] for this equivalence
class.
If b[a]R, b is called a representative of this equivalence
class.
Shiwani Gupta 53
54. Equivalence Classes
Example: In the previous example (strings of identical
length), what is the equivalence class of the word mouse,
denoted by [mouse] ?
Solution: [mouse] is the set of all English words containing
five letters.
For example, ‘horse’ would be a representative of this
equivalence class.
Shiwani Gupta 54
55. Partition of sets
❑ Example: Let A denote the set
of the lowercase English
alphabet. Let B be the set of
lowercase consonants and C
be the set of lowercase
vowels. Then B and C are
nonempty, B ∩ C = , and A
= B ∪ C. Thus, {B, C} is a
partition of A.
❑ Let A be a set and let {A1, A2,
A3, A4, A5} be a partition of A.
Corresponding to this
partition, a Venn diagram,
can be drawn, Figure 3.13
Shiwani Gupta 55
56. Equivalence Classes
Theorem: Let R be an equivalence relation on a set A. The
following statements are equivalent:
❑ aRb
❑ [a] = [b]
❑ [a] [b]
Definition: A partition of a set S is a collection of disjoint
nonempty subsets of S that have S as their union. In other words,
the collection of subsets Ai, iI, forms a partition of S if and only
if
(i) Ai for iI
❑ Ai Aj = , if i j
❑ iI Ai = S
Shiwani Gupta 56
57. Equivalence Classes
Examples: Let S be the set {u, m, b, r, o, c, k, s}.
Do the following collections of sets partition S ?
{{m, o, c, k}, {r, u, b, s}} yes.
{{c, o, m, b}, {u, s}, {r}} no (k is missing).
{{b, r, o, c, k}, {m, u, s, t}} no (t is not in S).
{{u, m, b, r, o, c, k, s}} yes.
{{b, o, o, k}, {r, u, m}, {c, s}} yes ({b,o,o,k} = {b,o,k}).
{{u, m, b}, {r, o, c, k, s}, } no ( not allowed).
Shiwani Gupta 57
58. Equivalence Classes
Theorem: Let R be an equivalence relation on a
set S. Then the equivalence classes of R form a partition of
S. Conversely, given a partition
{Ai | iI} of the set S, there is an equivalence relation R that
has the sets Ai, iI, as its equivalence classes.
Shiwani Gupta 58
59. Example: Equivalence Classes
Let us assume that Frank, Suzanne and George live in
Boston, Stephanie and Max live in Lübeck, and Jennifer
lives in Sydney.
Let R be the equivalence relation {(a, b) | a and b live in
the same city} on the set P = {Frank, Suzanne, George,
Stephanie, Max, Jennifer}.
Then R = {(Frank, Frank), (Frank, Suzanne),
(Frank, George), (Suzanne, Frank), (Suzanne, Suzanne),
(Suzanne, George), (George, Frank),
(George, Suzanne), (George, George), (Stephanie,
Stephanie), (Stephanie, Max), (Max, Stephanie),
(Max, Max), (Jennifer, Jennifer)}.
Shiwani Gupta 59
60. Equivalence Classes
Then the equivalence classes of R are:
{{Frank, Suzanne, George}, {Stephanie, Max}, {Jennifer}}.
This is a partition of P.
The equivalence classes of any equivalence relation R
defined on a set S constitute a partition of S, because every
element in S is assigned to exactly one of the equivalence
classes.
Shiwani Gupta 60
61. Example: Equivalence Classes
Let R be the relation
{(a, b) | a b (mod 3)} on the set of integers.
Is R an equivalence relation?
Yes, R is reflexive, symmetric, and transitive.
What are the equivalence classes of R ?
{{…, -6, -3, 0, 3, 6, …},
{…, -5, -2, 1, 4, 7, …},
{…, -4, -1, 2, 5, 8, …}}
Shiwani Gupta 61
62. ❑Quotient Set: A set denoted by A/R is the collection
of all distinct and disjoint sets of equivalence classes,
induced by an equivalence relation R
❑Quotient set is a partition of A
❑Circular Set: A relation R on set A is said to be
circular if aRb and bRc ⇒ cRa
Shiwani Gupta 62
63. Reflexive closure
Smallest reflexive relation R1 = Δ U R
❑Let R be a relation on set A and R is not reflexive then
a smallest reflexive relation on A is said to be reflexive
closure of R if it contains R
If R is a reflexive relation, then reflexive closure of R
is itself
Shiwani Gupta 63
64. Symmetric closure
Smallest symmetric relation R1 = R-1 U R
❑Let R be a relation on set A and R is not symmetric then
a smallest symmetric relation on A is said to be
symmetric closure of R if it contains R
If R is symmetric, then symmetric closure of R is itself
Shiwani Gupta 64
65. Transitive closure
Transitive closure of R = R ∞ i.e connectivity relation
❑Let R be a relation on set A and R is not transitive then a
smallest transitive relation on A is said to be transitive
closure of R if it contains R
If R is transitive, then transitive closure of R is itself
Transitive closure of non transitive relation can be found
by Warshall’s algorithm
Shiwani Gupta 65
69. Linearly Ordered Sets
Note: In a poset, every pair of elements need not be comparable
Eg. (R, ≤) is linearly ordered set whereas (Z+, |) is not
Shiwani Gupta 69
70. Partially Ordered Sets
❑ Digraphs of Posets
❑ Because any partial order is also a
relation, a digraph representation of
partial order may be given.
❑ Example: On the set S = {a, b, c},
consider the relation R =
{(a, a), (b, b), (c , c ), (a, b)}.
❑ From the directed graph it follows that
the given relation is reflexive and
transitive.
❑ This relation is also antisymmetric
because there is a directed edge from a
to b, but there is no directed edge from b
to a.
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71. Partially Ordered Sets
❑Digraphs of Posets
❑Let S = {1, 2, 3, 4, 6, 12}.
Consider the divisibility
relation on S, which is a
partial order
❑A digraph of this poset is as
shown in Figure 3.20
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73. Partially Ordered Sets
❑Closed Path
❑On the set S = {a, b, c } consider
the relation R = {(a, a), (b, b), (c ,
c ), (a, b), (b, c ), (c , a)}
❑The digraph of this relation is given
in Figure 3.21
❑In this digraph, a, b, c , a form a
closed path. Hence, the given
relation is not a partial order
relation
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74. Hasse Diagram
The digraph of a partial ordered relation can be
simplified and is called as Hasse Diagram.
When the partial order is a total order, its hasse diagram
is a straight line and the corresponding poset is called
chain
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76. Hasse Diagram
❑Let S = {1, 2, 3}. Then P(S)
= {, {1}, {2}, {3}, {1, 2},
{2, 3}, {1, 3}, S}
❑(P(S),≤) is a poset, where ≤
denotes the set inclusion
relation
❑Draw the digraph of this
inclusion relation (see Figure
3.23). Place the vertex A
above vertex B if B ⊂ A.
Now follow steps (2), (3),
and (4)
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77. ❑Let S = {1, 2, 3}. Then P(S) =
{, {1}, {2}, {3}, {1, 2}, {2, 3},
{1, 3}, S}
❑Now (P(S),≤) is a poset, where ≤
denotes the set inclusion relation.
The poset diagram of (P(S),≤) is
shown in Figure 3.22
Hasse Diagram
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78. ❑Quasi Order: A relation R on A is called quasi order if
it is transitive and irreflexive
❑Poset isomorphism: If f:A→A’ (one-one
correspondence) is an isomorphism then (A,≤) and
(A’,≤’) are known as isomorphic posets
A={1,2,3,5,6,10,15,30}
P(S)=A’={ϕ,{e},{f},{g},{e,f},{f,g},{e,g},{e,f,g}}
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79. Note: The greatest element of a poset is denoted by ‘I’ or’1’ is called
‘unit element’
The least element of a poset is denoted by 0 is called ‘zero
element’
Minimal and Maximal Elements
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80. ❑Consider the poset (S,≤), where S =
{2, 4, 5, 10, 15, 20} and the partial
order ≤ is the divisibility relation
❑In this poset, there is no element b ∈
S such that b 5 and b divides 5.
(That is, 5 is not divisible by any
other element of S except 5). Hence,
5 is a minimal element. Similarly, 2
is a minimal element
Hasse Diagram
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81. ❑10 is not a minimal element because
2 ∈ S and 2 divides 10. That is, there
exists an element b ∈ S such that b <
10. Similarly, 4, 15, and 20 are not
minimal elements
❑2 and 5 are the only minimal
elements of this poset. Notice that 2
does not divide 5. Therefore, it is not
true that 2 ≤ b, for all b ∈ S, and so 2
is not a least element in (S,≤).
Similarly, 5 is not a least element.
This poset has no least element
Hasse Diagram
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82. ❑There is no element b ∈ S such that b
15, b > 15, and 15 divides b. That is,
there is no element b ∈ S such that 15
< b. Thus, 15 is a maximal element.
Similarly, 20 is a maximal element.
❑10 is not a maximal element because
20 ∈ S and 10 divides 20. That is,
there exists an element b ∈ S such
that 10 < b. Similarly, 4 is not a
maximal element.
Figure 3.24
Hasse Diagram
Shiwani Gupta 82
83. ❑20 and 15 are the only
maximal elements of this
poset
❑10 does not divide 15, hence
it is not true that b ≤ 15, for
all b ∈ S, and so 15 is not a
greatest element in (S,≤)
❑This poset has no greatest
element
Figure 3.24
Hasse Diagram
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90. Lattice
A lattice L is said to be bounded if it has greatest element
1 and least element 0
If L is bounded lattice then for all a ∈ A, 0≤ a≤ 1
aV0=a, a ∧0=0, aV1=1, a ∧ 1=a
Let (L,≤) be the lattice, a nonempty subset S of L is called
sublattice of L if ∀ a,b ∈S, aVb, a ∧b ∈S
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91. ❑Non-distributive Lattice
❑Because a ∧ (b ∨ c ) = a ∧ 1 = a = 0
= 0 ∨ 0 = (a ∧ b) ∨ (a ∧ c ), this is
not a distributive lattice
Lattice
1
a
b
c
0
A Lattice is non distributive iff it contains
a sublattice which is isomorphic to one of
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92. Complemented Lattice
A Lattice is said to be complemented if it is bounded and
every element of L has a complement.
Complement of each element of bounded, distributive
lattice is unique.
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93. ❑Modular Lattice: A lattice is said to be modular if for
all a,b,c, a≤c→aV(b ∧c)=(aVb) ∧c
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95. Properties of Boolean Algebra
❑Every Boolean Algebra is isomorphic to the Boolean
Algebra (P(S), ⊆), where S is some set
❑Every Boolean Algebra must have elements of the
form 2n
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96. Application: Relational Database
❑In a relational database system, tables are considered
as relations
❑A table is an n-ary relation, where n is the number of
columns in the tables
❑The headings of the columns of a table are called
attributes, or fields, and each row is called a record
❑The domain of a field is the set of all (possible)
elements in that column
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97. n-ary Relations
In order to study an interesting application of relations,
namely databases, we first need to generalize the concept
of binary relations to n-ary relations.
Definition: Let A1, A2, …, An be sets. An n-ary relation
on these sets is a subset of A1A2…An.
The sets A1, A2, …, An are called the domains of the
relation, and n is called its degree.
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98. n-ary Relations
Example:
Let R = {(a, b, c) | a = 2b b = 2c with a, b, cN}
What is the degree of R?
The degree of R is 3, so its elements are triples.
What are its domains?
Its domains are all equal to the set of integers.
Is (2, 4, 8) in R?
No.
Is (4, 2, 1) in R?
Yes.
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99. Databases and Relations
Let us take a look at a type of database representation that is
based on relations, namely the relational data model.
A database consists of n-tuples called records, which are
made up of fields.
These fields are the entries of the n-tuples.
The relational data model represents a database as an n-ary
relation, that is, a set of records.
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100. Databases and Relations
Example: Consider a database of students, whose records
are represented as 4-tuples with the fields Student Name, ID
Number, Major, and GPA:
R = {(Ackermann, 231455, CS, 3.88),
(Adams, 888323, Physics, 3.45),
(Chou, 102147, CS, 3.79),
(Goodfriend, 453876, Math, 3.45),
(Rao, 678543, Math, 3.90),
(Stevens, 786576, Psych, 2.99)}
Relations that represent databases are also called tables,
since they are often displayed as tables.
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