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Ch. 8: Relations

Ch. 8: Relations. 8.1 Relations and their Properties. Functions. Recall ch . 1: Functions Def. of Function: f:A→B assigns a unique element of B to each element of A. Functions- Examples and Non-Examples. Ex: students and grades. Function Ex. Ex: A={1,2,3,4,5,6}, B={ a,b,c,d,e,f }

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Ch. 8: Relations

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  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: • 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 216 How many relations are there on a set with n elements? 2n^2 Number of reflexive relations on a set with n elements • The other n(n-1) may or may not be in. • So 2n(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 οR = {(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 οR Find R οS

  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 οR Find R οS

  24. Def of powers Def: Let R be a relation on the set A. The powers Rn, n=1,2,3,… are defined inductively by R1=R and Rn+1=Rn R

  25. Ex Ex: R={(1,1), (2,1), (3,2), (4,3)} R2= {(1,1), (2,1), (3,1), (4,2)} R3=… Show R4=R3 So Rn=R3 for n=4, ..

  26. Ex: R={(1,1), (1,2), (3,4), (4,5), (3,5)} R2 = {(1,1), (1,2), (3,5)} R3={(1,1), (1,2)} R4=R3 so Rn=R3

  27. Thm. 1 Theorem 1: Let R be a transitive relation on a set A. Then Rn 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 Rn 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.

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