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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 NonExamples Ex: students and grades 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. 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. Examples of relations • Flights 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 Functions and Relations • Do a few examples of students and grades and determine if they are functions and/or relations Notations for Relations Notations: • Graphical • Tabular • Ordered pairs • aRb • later: matrices and digraphs 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 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 Question • What does RST show? • RAT? 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 More examplesR on the set A of all people. 5. R={(a,b)| a’s last name starts with the same letter as b’s } RSAT 6. R={(a,b)| a is a (full) sister of b } RSAT Let A=set of subsets of a nonempty set 7. R={(a,b)| a is a subset of b } RSAT Let A={1,2,3,4} 8. R={(a,b)| a divides b } R={(1,1),(1,2),(1,3),(1,4),(2,2),…} RSAT 9. R={(1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4)} RSAT Let A=Z (integers) 10. R={(a,b)| a≤ b } RSAT 11. R={(a,b)| a=b+1 } RSAT 12. R={(1,1), (2,2), (3,3) } RSAT 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. 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. 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 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} 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 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 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 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, .. 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 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? 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.