Ch9.5-Equivalence-relations.ppt - Equivalence Relations...

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1 Equivalence Relations Jorge Cobb The University of Texas at Dallas
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2 Review questions What is the reflexive closure of the relation {( x , y ): x y } on the set of integers? Let R be the relation that contains a pair ( x , y ) if there is a direct flight from city x to city y . What is the meaning of ( x , y ) R 3 ? Apply the theorem discussed in class to find the transitive closure of the relation {(2, 1), (2, 3), (3, 1), (3, 4), (4, 1), (4, 3)} on the set {1, 2, 3, 4, 5}.
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3 Two major kinds of relations Equivalence relations reflexive symmetric transitive Partial orders reflexive antisymmetric transitive
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EQUIVALENCE RELATIONS 4
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5 Example equivalence relations The relation “=” on any set of numbers (integers, reals, or any subset thereof) What about relation R on the reals such that a R b iff a – b is integer. Reflexive? a R a? Symmetric? a R b b R a? Transitive? (a - b) integer, and (b - c) integer, (a – c) integer because (a – c) = (a – b) + (b – c), both are integers
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6 Another example The relation “ x = y (mod m )” on integers, where m > 1, or congruence modulo m . Recall that x = y (mod m ) iff m divides x y Reflexive? x x = 0 which any m divides Symmetric? If m divides x y and the divisor is k , then m divides y x with divisor k Transitive? If x y = km and y z = lm , then x z = ( x y ) + ( y z ) = km + lm = ( k + l ) m BTW, x = y (mod m ) iff (x mod m) = (y mod m)
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7 Other examples What about the “divides” relation on the set of integers? reflexive, transitive, but not symmetric What about the relation {( x , y ): | x y | < 1} with x and y real numbers? reflexive; | x x | = 0 < 1 symmetric; | x y | = | y x | not transitive; for example, x = 1, y = 1.8, z = 2.5
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8 Equivalence classes Two elements x and y that are related by an equivalence relation are called equivalent (with respect to that relation). We denote this by x y (also x R y for relation R) The set of all elements related to x is called the equivalence class of x , denoted [ x ] R or [ x ] if the relation is understood.
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  • Fall '19
  • Equivalence relation, representative, Transitive relation, Partially ordered set

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