Transcript 8.1 properties

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:
•
•
•
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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.