1. Reflexivity, Symmetry, and Transitivity Lecture 44 Section 10.2 Thu, Apr 7, 2005

2. 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.

3. Examples Which of the following relations are reflexive? symmetric? transitive? a  b, on Z. A  B, on  (U). p  q, on a set of statements. a  b (mod 10), on Z. gcd(a, b) > 1, on Z. p  q = p, on a set of statements.

4. Examples Which of the following relations are reflexive? symmetric? transitive? R  R, on R. , on R.

5. 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 R r defined by R r = R  {(a, a) | a  A}. Then R r is the “smallest” relation that includes R and is reflexive.

6. The Reflexive Closure of a Relation The reflexive closure of < (on R ) is . The reflexive closure of  is . The reflexive closure of a  b (mod 10) on Z is itself.

7. 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 R s defined by R s = R  {(a, b) | (b, a)  R}. Then R s is the “smallest” relation that includes R and is symmetric.

8. 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 ?

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

10. 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. While R n  R n – 1 Let R n + 1 = R n  {(a, c)  (a, b), (b, c)  R n for some a, b, c  A} n = n + 1 End While

11. 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 *.