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# lect10_1 - Todays topics Reflexive symmetric and transitive...

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1 Today’s topics: Reflexive, symmetric and transitive closures of relations Equivalence relations

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2 Closures A closure “extends” a relation to satisfy some property. But extends it as little as possible. Definition. The closure of relation R with respect to property P is the relation S that i) contains R ii) satisfies property P iii) is contained in any relation satisfying i) and ii). That is S is the “smallest” relation satisfying i) and ii).
3 Just a reminder: reflexive: aRa symmetric: aRb bRa transitive: aRb bRc aRc anti-symmetric: aRb bRa a = b , i. e. 2200 a b , aRb ¬ bRa

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4 Lemma 1. The reflexive closure of R is S = R {( a , a ) | a A } =r ( R ) . S contains R and is reflexive by design. Furthermore, any relation satisfying i) must contain R , any satisfying ii) must contain the pairs ( a , a ), so any relation satisfying both i) and ii) must contain S Proof. In accordance to the definition of a closure, we need to prove three things to show that S is reflexive closure: i) S contains R ii) S is reflexive iii) S is the smallest relation satisfying i) and ii).
5 Example: R ={( a , b ), ( a , c ), ( b , d ), ( d , e )} a b c d e a 0 1 1 0 0 b 0 0 0 1 0 c 0 0 0 0 0 d 0 0 0 0 1 e 0 0 0 0 0 R a b c d e a 1 1 1 0 0 b 0 1 0 1 0 c 0 0 1 0 0 d 0 0 0 1 1 e 0 0 0 0 1 S = r ( R ) S ={( a , b ), ( a , c ), ( b , d ), ( d , e ), ( a , a ), ( b , b ), ( c , c ), ( d , d ), ( e , e )}

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6 Sometimes the relation on A that consists of all loops is called identity relation on A , i. e. I A = {( a , a )| a A } Then reflexive closure of a relation R A × A is r ( R )= R I A Example of a reflexive closure. Let A be any set and consider the relation on Power( A ) R = {( x , y ) Power( A ) × Power( A ) | x y } The reflexive closure of R would be the relation: R I Power( A ) ={( x , y ) Power( A ) × Power( A )|( x , y ) R or ( x , y ) I Power( A ) } ={( x , y ) Power( A ) × Power( A )| x y
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lect10_1 - Todays topics Reflexive symmetric and transitive...

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