Equivalence Relations is an equivalence iff is

Equivalence Relations is an equivalence iff is

10 Pages · 1999 · 24 KB · English

Discrete Mathematics by Section 6.5 and Its Applications 4/E Kenneth Rosen TP 1 Section 6.5 Equivalence Relations Now we group properties of relations together to define

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Discrete MathematicsbySection 6.5 and Its Applications 4/EKenneth RosenTP 1 Section 6.5 Equivalence RelationsNow 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 and ? transitive _______ It is easy to recognize equivalence relations using digraphs. ? The subset of all elements related to a particular element forms a universal relation (contains all possible arcs) on that subset. The (sub)digraph representing the subset is called a complete (sub)digraph. All arcs are present. ? The number of such subsets is called the rank of the equivalence relation ______ Discrete MathematicsbySection 6.5 and Its Applications 4/EKenneth RosenTP 2 Examples: A has 3 elements:rank = 3rank = 2 Discrete MathematicsbySection 6.5 and Its Applications 4/EKenneth RosenTP 3rank = 2rank = 2rank = 1 Discrete MathematicsbySection 6.5 and Its Applications 4/EKenneth RosenTP 4 ___________ ? Each of the subsets is called an equivalence class. ? A bracket around an element means the equivalence class in which the element lies. [x] = {y | is in R} ? The element in the bracket is called a representative of the equivalence class. We could have chosen any one. ________ Examples:abc[a] = {a, c}, [c] = {a, c}, [b] = {b}. rank = 2 ________ An interesting counting problem: Discrete MathematicsbySection 6.5 and Its Applications 4/EKenneth RosenTP 5 Count the number of equivalence relations on a set A with n elements. Can you find a recurrence relation? The answers are ? 1 for n = 1 ? 3 for n = 2 ? 5 for n = 3 How many forn = 4? ____________ Definition: Let S 1, S 2, . . ., S n be a collection of subsets of A. Then the collection forms a partition of A if the subsets are nonempty, disjoint and exhaust A: ? S i ¹ Æ? S iÇS j = Æ if i ¹ j ? U S i = A Discrete MathematicsbySection 6.5 and Its Applications 4/EKenneth RosenTP 6ASSS234SS15___________ Theorem: The equivalence classes of an equivalence relation R partition

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