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

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 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? 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 RelationsEx: 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

RS

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 exampleEx 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 trueInductive 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.