Ch 8 Relations 8 1 Relations and their

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} {(1, a), (2, c), (3, b), (4, f), (5, b), (6, c)} is a subset of Ax. B Also show graphical format.

Relations are also subsets of Ax. B, 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 Ax. B. Special Case: A relation on the set A is a relation from A to A.

Examples of relations • Flights

Review of Ax. B • Recall that Ax. B={(a, b)|a � A and b � B} • For A={1, 2, 3} and B={x, y}, find Ax. B • Find Ax. A

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 • a. Rb • 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 a. Ra for every a � A symmetric if b. Ra whenever a. Rb for every a, b � A antisymmetric : a. Rb and b. Ra only if a=b for a, b � A transitive if whenever a. Rb and b. Rc, then a. Rc 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: 1. R={(a, b)| a is older than b } circle if so: 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 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

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

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

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

Number of relations-questions How many relations are there on a set with 4 elements? Ax. A 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? Ax. A has 4^2=16 elements. So number of subsets is 216 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.

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, 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 R 1=R and Rn+1=Rn R

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 Rn=R 3 for n=4, . .

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 Rn=R 3

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.