Reflexivity Symmetry and Transitivity Lecture 44 Section 10

Reflexivity, Symmetry, and Transitivity Lecture 44 Section 10. 2 Thu, Apr 7, 2005

Reflexivity, Symmetry, and Transitivity Let R be a relation on a set A. ¢ R is reflexive if x A, (x, x) R. ¢ R is symmetric if x, y A, (x, y) R (y, x) R. ¢ R is transitive if x, y, z A, (x, y) R and (y, z) R (x, z) R. ¢

Examples ¢ Which of the following relations are reflexive? symmetric? transitive? a b, on Z. l A B, on (U). l p q, on a set of statements. l a b (mod 10), on Z. l gcd(a, b) > 1, on Z. l p q = p, on a set of statements. l

Examples ¢ Which of the following relations are reflexive? symmetric? transitive? R R, on R. l

The Reflexive Closure of a Relation Let R be a relation on a set A. ¢ Define the reflexive closure of R to be the relation Rr defined by Rr = R {(a, a) | a A}. ¢ Then Rr is the “smallest” relation that includes R and is reflexive. ¢

The Reflexive Closure of a Relation The reflexive closure of < (on R) is . ¢ The reflexive closure of a b (mod 10) on Z is itself. ¢

The Symmetric Closure of a Relation Let R be a relation on a set A. ¢ Define the symmetric closure of R to be the relation Rs defined by Rs = R {(a, b) | (b, a) R}. ¢ Then Rs is the “smallest” relation that includes R and is symmetric. ¢

The Symmetric Closure of a Relation What is the symmetric closure of a b, on Z? ¢ What is the symmetric closure of gcd(a, b) > 1, on Z? ¢

The Transitive Closure of a Relation Let R be a relation on a set A. ¢ The transitive closure of R is a relation Rt with the following properties. ¢ Rt is transitive. l R Rt. l If S is a transitive relation on A and R S, then Rt S, i. e. , Rt is the “smallest” transitive relation that contains R. l

Finding the Transitive Closure of a Relation ¢ Given a relation R on a set A, the following algorithm will produce the transitive closure of R. Let n = 1 and R 0 = and R 1 = R. l While Rn – 1 l • Let Rn + 1 = Rn {(a, c) (a, b), (b, c) Rn for some a, b, c A} • n=n+1 l End While

Closures of a Relation Let P be the set of all polynomials with real coefficients. ¢ Define the relation R on P by (f, g) R if df/dx = g. ¢ What is the reflexive closure of R? ¢ What is the symmetric closure of R? ¢ What is the transitive closure of R? ¢ What is the reflexive-transitive closure of R? Call it R*. ¢