Chapter 8 Equivalence Relations

Aug 20, 2014

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Chapter 8 Equivalence Relations. Let A and B be two sets. A r elation R from A to B is a subset of AXB. That is, R is a set of ordered pairs, where the first coordinate of the pair belongs to A and the second coordinate belongs to B.

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Chapter 8 Equivalence Relations

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Equivalence Relations

Equivalence Relations. Rosen 6.5. Some preliminaries. Let a be an integer and m be a positive integer. We denote by a MOD m the remainder when a is divided by m. If r = a MOD m, then a = qm + r and 0  r  m, qZ Examples Let a = 12 and m = 5, 12MOD5 = 2

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Rosen, Section 8.5 Equivalence Relations

Rosen, Section 8.5 Equivalence Relations. Longin Jan Latecki Temple University, Philadelphia latecki@temple.edu Some slides from Aaron Bloomfield. Introduction. Certain combinations of relation properties are very useful

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Ordered Sets. Relations. Equivalence Relations

Ordered Sets. Relations. Equivalence Relations. Ordered Pair. When listing the elements of a set, the order in which the elements is immaterial. In ordered pair of elements a and b, denoted (a, b), the order in which the entries are written is taken into account.

624 views • 18 slides

Chapter 8: Recurrence Relations

Chapter 8: Recurrence Relations. Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about recurrence relations Learn the relationship between sequences and recurrence relations Explore how to solve recurrence relations by iteration

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8.5 Equivalence Relations

8.5 Equivalence Relations. Now we group properties of relations together to define new types of important relations. Definition : A relation R on a set A is an equivalence relation iff R is • reflexive • symmetric • transitive. Equivalence Relations.

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Equivalence Relations

Equivalence Relations . Equivalence relations are used to relate objects that are similar in some way. Definition: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

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8.5 Equivalence Relations

8.5 Equivalence Relations. Def. of Equivalence Relation. Def: A relation on a set A is called an equivalence relation if it is R, S, and T (Reflexive, Symmetric, and Transitive).

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Equivalence Relations

Equivalence Relations. Equivalence Relations. A relation R on a set A is an equivalence relation if it is reflexive, symmetric, and transitive a~b denotes that a and b ( a,b ∈ A) are equivalent elements with respect to R

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Section 7.5: Equivalence Relations

Section 7.5: Equivalence Relations. Def : A relation R on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. Ex : Let A be any set and define R = {(a, b) | a = b}. That is, every element of A is related to itself only.

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Equivalence Relations

Equivalence Relations. Equivalence Relations. A relation R on A A is an equivalence relation when R satisfies 3 conditions:  x  A, xRx ( reflexive ).  x, y  A, xRy  yRx ( symmetric ).  x, y, z  A, (xRy  yRz)  xRz ( transitive ).

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Equivalence Relations

Equivalence Relations. Rosen 7.5. Equivalence Relation. A relation on a set A is called an equivalence relation if it is Reflexive Symmetric Transitive Two elements that are related by an equivalence relation are called equivalent. Some preliminaries.

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Equivalence Relations

Equivalence Relations. Aaron Bloomfield CS 202 Rosen, section 7.5. Introduction. Certain combinations of relation properties are very useful We won’t have a chance to see many applications in this course In this set we will study equivalence relations

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Equivalence Relations

Equivalence Relations. Fractions vs. Rationals. Question: Are 1/2, 2/4, 3/6, 4/8, 5/10, … the same or different? Answer: They are different symbols that stand for the same rational number.

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Chapter 8: Relations

Chapter 8: Relations. Relations(8.1) n-any Relations & their Applications (8.2). Relations (8.1). Introduction Relationship between a program and its variables Integers that are congruent modulo k Pairs of cities linked by airline flights in a network. Relations (8.1) (cont.).

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Chapter 8 Investor Relations

Chapter 8 Investor Relations. 8- 1. Definition of Investor Relations. The National Investor Relations Institute defines Investor Relations as:

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Chapter 8 Relations

歐亞書局. Discrete Mathematics and Its Applications Sixth Edition By Kenneth Rosen. Chapter 8 Relations. 歐亞書局. 8.1 Relations and Their Properties 8.2 n-ary Relations and Their Applications 8.3 Representing Relations 8.4 Closures of Relations 8.5 Equivalence Relations

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Equivalence Relations

Equivalence Relations. A relation on set A is called an equivalence relation if it is: reflexive, symmetric, and Transitive We use the notation a  b. Example. Let R be a relation on set A , where A = {1, 2, 3, 4, 5} and R = {(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,1)}

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Discrete Mathematics Equivalence Relations

Discrete Mathematics Equivalence Relations. Introduction. Certain combinations of relation properties are very useful We won’t have a chance to see many applications in this course In this set we will study equivalence relations A relation that is reflexive, symmetric and transitive.

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Section 7.5 Equivalence Relations

Section 7.5 Equivalence Relations. Longin Jan Latecki Temple University, Philadelphia latecki@temple.edu. We can group properties of relations together to define new types of important relations. _________________

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Section 7.5 Equivalence Relations

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Equivalence Relations

Equivalence Relations. MSU CSE 260. Outline. Introduction Equivalence Relations Definition, Examples Equivalence Classes Definition Equivalence Classes and Partitions Theorems Example. Introduction. Consider the relation R on the set of MSU students:

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Equivalence Relations

Equivalence Relations. Lecture 45 Section 10.3 Fri, Apr 8, 2005. Equivalence Relations. An equivalence relation on a set A is a relation on A that is reflexive, symmetric, and transitive. We often use the symbol ~ as a generic symbol for an equivalence relation.

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