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Unformatted text preview: CSE 1400 Applied Discrete Mathematics Relations Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Relations and Their Graphs 1 A Relation’s Domain, Codomain, and Range 2 A Sampling of Relations 3 Equality 3 Less than 3 Divides 3 Congruence Modulo n 4 Perpendicular on Lines 5 The Incestuous and Empty Relations 5 A Relation is a Set of Ordered Pairs 6 The Inverse Relation 6 Counting Relations 7 Relational Properties 7 Reflexive Property 7 Symmetric Property 8 Antisymmetric Property 10 Transitive Property 11 Orders and Equivalences 11 Orders 12 WellOrdered Sets 13 Equivalences 13 Equivalence Relations Partition a Set 14 Stirling Numbers of the Second Kind 16 cse 1400 applied discrete mathematics relations 2 Problems on Relations 18 Abstract A relation ∼ describes how things are connected. That a thing a is related to a thing b can be represented by 1 . An ordered pair ( a , b ) . 2 . An directed edge a • • b . 3 . Or more commonly, simply using relational notation a ∼ b . A relation is 1 . A set G of ordered pairs . 2 . A directed graph G of nodes and edges. 3 . A matrix of True and False values. Higherdimensional relations among a , b , c or more parameters can be defined. Higherdimensional relations occur as tables in relational databases and as data in multivariable problems. Relations and Their Graphs A relation is a set of ordered pairs . A relation can be pictured of as a graph G . • a • b c • d • • e • f h • • g This graph represents the relation G = { ( a , b ) , ( b , c ) , ( c , d ) , ( d , h ) , ( h , f ) , ( f , e ) , ( e , a ) } G = { ( x , y ) : x ∼ y } = { ( x , y ) : x is related to y . } There are many examples of relations. You are, no doubt, familiar with relations among people: MotherDaughter, FatherSon, Parent Child, AuntNephew. Familial relations often become murky. We study relations that can be precisely defined. A few common rela tions are equality, congruence mod n , less than, divides, subset, and perpendicular. In this course, relationships will be between two things a and b . Relationships among 3 or more things are common and useful, but The course studies binary relations. these ideas are not within the scope of this course. A Relation’s Domain, Codomain, and Range T he things involved in a relation need names : C all them the things x and y . Write x ∼ y to express the phrase “ x is related to y .” cse 1400 applied discrete mathematics relations 3 The value x belongs to a set X called the domain of ∼ . The value y belongs to a set Y called the co domain of ∼ . The domain X is the set of elements that appear on the lefthand side of ∼ . For this course, you can assume that every element in X appears on the lefthand side of ∼ , that is, every relation we en counter is total , defined on all members of X . A relation ∼ is said to be total when every x ∈ X occurs on on the lefthand side of ∼ for some y ∈ Y ....
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 Fall '11
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