1. Chap6 Relations Def 1: Let A and B be sets. A binary relation from A
to B is a subset of A x B
a R b: (a,b) R
a R b: (a,b)  R
Figure1
Def 2: A relation on the set A is a relation from A
to A
Example 4
Example 6 ( How many “possible“ relations…)

2. Chap6 Relations Def 3: A relation R on a set A is called reflexive
if (a,a) R for every element a A .
Example 7

3. Chap6 Relations Def 4:A relation R on a set A is called symmetric
if (b,a) R whenever (a,b) R , for a,b A.
A relation R on a set A such that (a,b) R
and (b ,a) R only if a=b , for a,b A , is
called antisymmetric .
Example 10
a relation can be both or neither
ex both : {(1,1)}
neither : { (1,2), (2,3),(3,2) }

4. Chap6 Relations Def 5: A relation R on a set A is called transitive
if whenever (a,b) R and (b,c) R ,then
(a,c) R , for a,b,c A.
Example 13
Example 16

5. Chap6 Relations Def 6:Let R be a relation from a set A to a set B and
S be a relation from B to a set C . The
composite of R and S is the relation consisting
of ordered pairs (a,c) , where a A ,cC ,
and for which there exists an element b  B
such that (a,b) R and (b,c)  S.
We denote the composite of R and S by S  R
Example 19

6. Chap6 Relations Def 7 : Let R be a relation on a set A . The power 3 4
Rn , n=1,2,3,…, are defined inductively
by R1=R and R n+1=Rn  R
Example 20

7. Theorem1: The relation R on a set A is transitive
if and only if Rn  R , for n=1,2,3,…..
“ if “ Rn  R , R2  R
if ( a,b)  R and ( b,c )  R ,( a,c ) R2
R2  R => (a,c)  R => R is transitive

8. Chap6 Relations assume (a,b)  Rk+1 since Rk+1 = Rk  R
“ only if “ by mathematical induction
n=1,trivial
assume Rk  R to show Rk+1  R
assume (a,b)  Rk+1 since Rk+1 = Rk  R
there exist x A such that
(a,x)  R and ( x,b)  Rk
since Rk  R , ( x,b)  R
( a,b)  R
Rk+1 R

9. Chap6 Representing Relations Using Digraphs
Def 1:
-Example 7
-determine whether a relation is reflexive ,
symmetric ,antisymmetric and transitive
-Example 10

10. Chap6 Closure of Relation
R= { (1,1), (1,2) , (2,1), (3,2) } on A = {1,2,3} is not reflexive
R’ = { (1,1), (1,2) , (2,1), (3,2) , (2,2), (3,3)}
R  R’ , R’ is reflexive , R’  , where is any reflexive relation and R 
R’ is the reflexive closure of R
R’ = R  , ={(a,a)| a A} is the
diagonal relation on A
Example 1 R R R

11. Chap6 Closure of Relation
R= { (1,1), (1,2) , (2,2), (2,3) , (3,1), (3,2)} on {1,2,3}
R’= { (1,1), (1,2) ,(2,1), (2,2), (2,3) ,(3,1), (3,2),(1,3)}
R  R’ , R’ is symmetric , R’  , where
is any symmetric relation and R 
R’ is the symmetric closure of R
R’=R  R-1, R-1={(b,a) | ( a,b) R, a=b}
Example 2 R R R

12. Chap6 Closure of Relation
-R= { (1,3), (1,4) , (2,1) , (3,2)} on {1,2,3,4}
add (2,4), (3,1) ,(2,3) ,(1,2) to become transitive?

13. Chap6 Relations Def 1: path ,cycle Theorem 1
Example 3
Theorem 1
Let R be a relation on a set A .There is a path of length n from a to b if and only if ( a,b)  Rn

14. Chap6 Relations Def 2: Let R be a relation on a set A . The
connectivity relation R* consists of the pairs
(a,b) such that there is a path between a and
b in R
R* =  Rn
Example 4 a Rb if a has met b
R2 contains ( a,b) if  c,( a,c) R and ( c,b) R
Rn contains ( a,b) if  x1,x2,..,x n-1 ,( a,x1) R , (x1,x2) R ,…(x n-1,b) R
R*, dose R* contain ( you, Mongolis presidat ) ? 
n=1

15. Chap6 Relations Theorem 2 The transitive closure of a relation R equal
the connectivity relation R*
Proof: R  R*
have to show : 1. R* is transitive
2. R*  where is
transitive and R  R R R

16. Chap6 Relations ( a,c) R* R* is transitive
If ( a,b) R* and (b,c) R* ,then there are paths from a to b and from b to c in R
( a,c) R*
R* is transitive b c a a

17. Chap6 Relations since * =  k and k  , * 
is transitive and R  to prove R* 
n is also transitive and n 
since * =  k and k  , * 
since R  , R*  * ( any path in R is also a
path in )
R*  * 
R* = R  R2  R3  …..  Rn R R R R R R  R R R R R R
k=1 R R R R R

18. Chap6 Relations
xj+1
a=x1 x2
xi-1
xi =xj
b=xm
xi+1
xj-1
xj-2
m>n

19. Chap6 Equivalence Relations
Def 1 :A relation on a set A is an equivalence
relation if it is reflexive, symmetric, and
transitive
Two elements related by an equivalence relation are called equivalent.
Example 1
R: relation on the set of strings of English
letters such that aRb iff l (a)=l (b), where
l(x) is the length of the string X. Is R an
equivalence relation ?

20. Chap6 Equivalence Relations
l(a) =l(a) : reflexive
Suppose a R b , l(a)= l(b) , then since
l(b)=l(a), b R a : symmetric
suppose a R b and b R c , l(a)= l(b) and l(b)=l(c) ,
l(a)=l(c), a R c : transitive

21. Chap6 Equivalence Relations
Example 2 R: relation on the set of real numbers,
aRb iff a-b is an integer .
transitive : aRb and bRc, a - b and b-c are
integers a-c=(a-b)+(b-c) is an integer
Example 3 Congruence Modulo m
show R={(a,b)|ab(mod m)} is an
equivalence relation on the set of integers
, m ﹥1

22. Chap6 Equivalence Relations
a  b ( mod m ) iff m divides a – b
– a – a = 0 is divisible by m
– a – b is divisible by m, a – b = km
b – a = ( – k )m, b  a ( mod m )
– a  b ( mod m ) and b  c ( mod m )
m divides a – b and b – c
a – b = km and b – c = lm
a – c = ( a – b ) + ( b – c ) = ( k+l )m

23. Chap6 Equivalence Relations
Equivalence classes
A: set of students at NDHU, who graduated
from high school
R on A:{(x,y)|x and y graduated from same
high school }
given a student x, we can form a set of all
students equivalent to x with respect to R ;
This set is an equivalence class of R, denoted [x]R
[x]R = { s | ( x , s )  R }

24. Chap6 Equivalence Relations
Example 6 What are the equivalence classes of
0 and 1 for congruence modulo 4 ?
0 : a  0 (mod 4), integers divisible by 4
[0]= { …,-8 ,-4, 0, 4, 8,…}
1: a 1 (mod 4) a-1 divisible by 4
integers having a remained 1
when divided by 4

25. Chap6 Equivalence Relations
[1]= { …,-7 ,-3, 1, 5, 9,…}
The equivalence classes of the relation
congruence modulo m are congruence
classes modulo m
[a]m={…, a-2m, a-m, a, a+m, a+2m,…}
x  a (mod m)
x – a = km
x = a + km

26. Chap6 Equivalence Relations
Partitions
A: Students majoring in exactly one subject
R on A:{(x ,y)|x and y have same major }
R splits all students in A into a collections
of disjoint subsets, where each subset
contains students with a specified major

27. Chap6 Equivalence Relations
Theorem 1. Let R be an equivalence relation on
a set A. The following statements
are equivalent:
( i ) a R b
(ii ) [ a ] = [ b ]
(iii) [ a ]  [ b ]  

28. Chap6 Equivalence Relations
Proof: (i) implies (ii)
a R b, prove [a]=[b], to show [a]  [b] and
[b]  [a]
suppose c  [a], to show c  [b]
a R c, since a R b, b R a  b R c c  [b]
(ii) implies (iii) [a] is nonempty,
[a][b]

29. Chap6 Equivalence Relations
(iii)implies (i)
[a][b], there is an element c ,
c  [a] and c  [b]
a R c and b R c  a R b
We have shown (i) implies (ii), (ii) implies
(iii) , and (iii) implies (i) . (i) , (ii) , (iii)
are equivalent

30. Chap6 Equivalence Relations
a partition of a set S is a collection of disjoint
nonempty subsets of S that have S as their union
the collection of subsets Ai , iI forms a
partition of S iff
Ai   for iI
Ai  Aj , when ij,
and
 Ai= S
iI
Figure 1

31. Chap6 Equivalence Relations
an equivalent relation partitions a set
 [ a ]R= A
aA
[a]R  [b]R= when [a]R  [b]R

32. Chap6 Equivalence Relations
Theorem 2 Let R be an equivalence on a set S.
Then the equivalence classes of R
form a partition of S. Conversely,
given a partition {Ai | iI }of the
set S, there is an equivalence
relation R that has the set Ai, iI,
as its equivalence classes

33. Chap6 Equivalence Relations
There are m different congruence classes
modulo m , corresponding to the m different
remainders possible when an integer is
divided by m
[ 0 ]m, [ 1 ]m,…, [ m-1]m

34. Chap6 Equivalence Relations
Example 8 What are the sets in the partition of
the integers arising from congruence
modulo 4 ?
[ 0 ]4={… , -8 , -4 , 0 , 4 , 8 , … }
[ 1 ]4={… , -7 , -3 , 1 , 5 , 9 , … }
[ 2 ]4={… , -6 , -2 , 2 , 6 , 10 , … }
[ 3 ]4={… , -5 , -1 , 3 , 7 , 11 , … }