lect10 - Closures of relations A closure extends a relation...

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1 Closures of relations 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).
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2 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).
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3 Lemma 2 . The symmetric closure of R is S = R R -1 = s ( R ) Proof . S is symmetric and contains R . It is also the smallest such . For suppose we have some symmetric relation T with R T . To show that T contains S we need to show that R -1 T . Take any ( a , b ) R -1 , it implies ( b , a ) R , and ( b , a ) T, since R T. But then ( a , b ) T as well, because T is symmetric by assumption. So, S = R R -1 T .
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4 Theorem. The transitive closure of a relation R is the set S = {( a , b ) | there is a path from a to b in R }= t ( R ) Proof. Let’s show that S is transitive. Suppose ( a , b ) S and ( b , c ) S. This means there is a path from a to b and a path from b to c in R. If we concatenate” them (attach the end of the ( a , b )-path to the start of the ( b , c )-path) we get a path from a to c . Thus, ( a , c ) S and S is transitive. We need to show, that any transitive relation T containing R contains S . Let’s prove by contradiction. Assume that there exists a transitive relation T containing R , that is smaller then S .
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5 More formally: assume by the way of contradiction that there exists a transitive relation T , such that R T , but S T. We want to show that it results to contradiction, that will prove the claim, that S is the smallest transitive relation, that includes R. / 5 ( x , y ) S but ( x , y ) T 5 ‘a path’ from x to y in R x y R x y since R T and T is transitive ( x , y ) T (contradiction) / S T
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6 An alternative way to represent transitive closure is as the union of paths with different length t ( R ) = S 1 S 2 S 3 where S n = {( a , b ) | there is a path of length n from a to b in R } Lemma 3 . The transitive closure of the relation R is t ( R ) = R R 2 R 3 (left for you to prove) Theorem . ( a , b ) R n iff there is a path of length n in R . (will be proved later by induction) Thus, we can prove the following equivalent definition of transitive closure
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7 Theorem . Suppose R is a relation on a set A , R A × A , | A |= n , and there is a path of length m > n from a to b in R.
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lect10 - Closures of relations A closure extends a relation...

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