cs340notes08Set - Chapter 8 Disjoint Set Class CS 340...

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Unformatted text preview: Chapter 8 Disjoint Set Class CS 340 Chapter 8: Disjoint Set Class 1 Relations A relation R on a set S maps every pair of elements in S to either TRUE or FALSE. For example, the greater than relation, >, is TRUE for the integer pair (5,3) (i.e., 5 > 3) but not for the pair (6,8) (since 6>8 is FALSE). S = (3, 5, 6, 8) F T T T 8 F F T T 6 F F F T 5 F F F F 3 8 6 5 3 > (5,3) 5 R 3 = True, where R is the relation 5 is related to 3 CS 340 Chapter 8: Disjoint Set Class 2 Equivalence Relations An equivalence relation R is a relation satisfying the following three properties: ( Reflexive ) R for every in set S. ( Symmetric ) For every pair and in set S, if R , then R . ( Transitive ) For every triple , , in set S, if R and R , then R . For example The greater than relation is not an equivalence relation for integers (since it lacks the symmetric property) S = (3, 5, 6, 8) F T T T 8 F F T T 6 F F F T 5 F F F F 3 8 6 5 3 > F T T T 8 F F T T 6 F F F T 5 F F F F 3 8 6 5 3 > R e f l e x i v e S y m m e t r i c CS 340 Chapter 8: Disjoint Set Class 3 More Equivalence Relation Examples An equivalence relation R is a relation satisfying the following three properties: ( Reflexive ) R for every in set S. ( Symmetric ) For every pair and in set S, if R , then R . ( Transitive ) For every triple , , in set S, if R and R , then R . For example Modulo-10 equality is an equivalence relation for integers The does not equal relation is not an equivalence relation for integers (since it lacks both the reflexive and transitive properties) The electrical connectivity relation is an equivalence relation for network system components (i.e., end-stations, servers, switches) The electrical connectivity (or just connectivity) relation is the one we will focus on in this chapter....
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cs340notes08Set - Chapter 8 Disjoint Set Class CS 340...

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