1. Ch. 8: Relations 8.1 Relations and their Properties

2. Functions Recall ch. 1: Functions Def. of Function: f:A→B assigns a unique element of B to each element of A

3. Functions- Examples and Non- Examples Ex: students and grades

4. Function Ex Ex: A={1,2,3,4,5,6}, B={a,b,c,d,e,f} {(1,a),(2,c),(3,b),(4,f),(5,b),(6,c)} is a subset of AxB Also show graphical format.

5. Relations Relations are also subsets of AxB, without the above uniqueness requirement of functions. Def. of Relations: Let A and B be sets. A binary relation from A to B is a subset of AxB. Special Case: A relation on the set A is a relation from A to A.

6. Examples of relations Flights

7. Review of AxB Recall that AxB={(a,b)|a  A and b  B} For A={1,2,3} and B={x,y}, find AxB Find AxA

8. Functions and Relations Do a few examples of students and grades and determine if they are functions and/or relations

9. Notations for Relations Notations: Graphical Tabular Ordered pairs aRb later: matrices and digraphs

10. Properties for a relation A relation R on a set A is called: reflexive if (a,a)  R for every a  A symmetric if (b,a)  R whenever (a,b)  R for a,b  A antisymmetric : (a,b)  R and (b,a)  R only if a=b for a,b  A transitive if whenever (a,b)  R and (b,c)  R, then (a,c)  R for a,b,c  A

11. Alternative notation A relation R on a set A is called: reflexive if aRa for every a  A symmetric if bRa whenever aRb for every a,b  A antisymmetric : aRb and bRa only if a=b for a,b  A transitive if whenever aRb and bRc, then aRc for every a, b, c  A

12. Question What does RST show? RAT?

13. Ex: Consider the following relations R on the set A of all people. Determine which properties (RSAT) hold:circle if so: 1.R={(a,b)| a is older than b } RSAT 2. R={(a,b)| a lives within 10 miles of b } RSAT 3. R={(a,b)| a is a cousin of b } RSAT 4. R={(a,b)| a has the same last name as b } RSAT

14. More examples- R on the set A of all people. 5. R={(a,b)| a’s last name starts with the same letter as b’s } R S A T 6. R={(a,b)| a is a (full) sister of b } R S A T

15. Let A=set of subsets of a nonempty set 7. R={(a,b)| a is a subset of b } R S A T

16. Let A={1,2,3,4} 8. R={(a,b)| a divides b } R={(1,1),(1,2),(1,3),(1,4),(2,2),…} R S A T 9. R={(1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4)} R S A T

17. Let A=Z (integers) 10. R={(a,b)| a≤ b } R S A T 11. R={(a,b)| a=b+1 } R S A T 12. R={(1,1), (2,2), (3,3) } R S A T

18. Number of relations-questions How many relations are there on a set with 4 elements? AxA has ___ elements. So number of subsets is ___ How many relations are there on a set with n elements? ___ Number of reflexive relations on a set with n elements The other ___may or may not be in. So ___ reflexive relations.

19. Number of relations- Answers How many relations are there on a set with 4 elements? AxA has 4^2=16 elements. So number of subsets is 2 16 How many relations are there on a set with n elements? 2 n^2 Number of reflexive relations on a set with n elements The other n(n-1) may or may not be in. So 2 n(n-1) reflexive relations.

20. Combining Relations Ex: sets A={1,2,3}, B={1,2,3,4}; Relations: R={(1,1),(2,2), (3,3)}, S={(1,1), (1,2), (1,3), (1,4)} R∩S R  S R – S S – R

21. Def. of Composite Let R be a relations from A to B and S a relations from B to C. The composite of R and S: S ο R = {(a,c)| a  A, c  C, and there exists b  B such that (a,b)  R and (b,c)  S}

22. Composite example Ex 1: R from {0,1,2,3,4} to {0,1,2,3,4}, S from {0,1,2,3,4} to {0,1,2,3,4} R={(1,0), (1,1), (2,1), (2,2), (3,0), (3,1)} S={(1,0), (2,0), (3,1), (3,2), (4,1)} Find S ο R Find R ο S

23. Ex 2 Ex. 2: R and S on the set of all people: Let R={(a,b)| a is the mother of b} S={(a,b)|a is the spouse of b} Find S ο R Find R ο S

24. Def of powers Def: Let R be a relation on the set A. The powers R n, n=1,2,3,… are defined inductively by R 1 =R and R n+1 =R n R

25. Ex Ex:R={(1,1), (2,1), (3,2), (4,3)} R 2 = {(1,1), (2,1), (3,1), (4,2)} R 3 =… Show R 4 =R 3 So R n =R 3 for n=4,..

26. Ex: R={(1,1), (1,2), (3,4), (4,5), (3,5)} R 2 = {(1,1), (1,2), (3,5)} R 3 ={(1,1), (1,2)} R 4 =R 3 so R n =R 3

27. Thm. 1 Theorem 1: Let R be a transitive relation on a set A. Then R n is a subset of R for n=1,2,3,… Proof— what method would work well?

28. Proof By Induction: N=1: trivially true Inductive Step: Assume R n  R where n  Z+. Show: _______ Assume (a,b)  R n+1. (Question: Show?____) Then, since R n+1 = R n ο R, ______________ Since ______, then ____  R. Since _____________ then ______  R.