(1) The document discusses binary relations and concepts such as ordered pairs, reflexive relations, symmetric relations, inverse relations, and representing relations using matrices. (2) It also covers topics like equivalence relations, equivalence classes, and partitioning a set based on an equivalence relation. (3) Examples are provided to illustrate key concepts like inverse relations, representing relations as directed graphs, and determining equivalence classes.

1. 1
Chapter 4
Relations
9/5/2013

2. 2
Relations
•
A B, ordered
pairs A
B
binary relation
• : A B binary relation A
B A B
• binary relation R R A B
aRb (a, b) R a R b
(a, b) R
A B = {a, b}
R = {( , a) , ( , b) , ( , a) , ( , b) }9/5/2013

3. 3
Relations
• (a, b) R a
b R
• : P , C
, D
•P = { , , , },
•C = { , , }
•D = {( , ), ( , ),
( , ), ( , )}
•
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4. 4
(Complementary Relations)
• R:A B binary relation
R :≡ {(a,b) | (a,b) R}
• , R:A B, R,
binary relation
R :≡ {(a,b) | (a,b) R} = (A B) − R
• R
U = A B
• R R < = {(a,b) |
a<b)}
: < = {(a,b) | (a,b) <} = {(a,b) | ¬(a<b)} = ≥
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5. 5
(Inverse
Relations)
• R:A B
R−1:B A
R−1 :≡ {(b, a) | (a,b) R}
, <−1 = {(b, a) | a<b} = {(b, a) | b>a} = >
• , R: →
a R b a b R :≡ {(a, b) | a b}
:
b R−1 a b a R−1 :≡ {(b, a) | b
a }
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6. 6
(Inverse
Relations)
Q: R N : xRy y = x 2
R :≡ {(x, y) | y = x 2} R -1
?
A: xRy y = x 2
R R -1 :
:
yR -1x y = x 2 R−1 :≡ {(y, x) | y = x
2}
xR -1y x = y 2 R−1 :≡ {(x, y) | x = y
2}
xR -1y y = ± x R−1 :≡ {(x, y) | y =9/5/2013

7. 7
Example
• R={(a,b) | a b}
• R-1 ={(b,a) | a b}
={(a,b) | b a}
• S={(x,y) R R | y=(x+1)/3}
• S-1 ={(y,x) R R | y=(x+1)/3}
={(x,y) R R | x=(y+1)/3}
={(x,y) R R | y=3x-1}
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8. 8
• A
A
• “<”
N
• identity relation IA A
{(a,a)|a A}
• : A = {1, 2, 3, 4}
R = {(a, b) | a < b} ?
•Solution: R = {(1, 2),
(1, 3), (1, 4), (2, 3),
(2, 4), (3, 4)}
1 1
2
3
4
2
3
4
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9. 9
•
A n ?
• A A A
• A A ?
• n2 A A, (=
A) A A ?
• m 2m
A A 2n2
•Answer: A
2n2
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10. 10
• : R A
reflexive (a, a) R
a A
• U =
• :
– < reflexive x < x ,
(x, x) R
– ≥ :≡ {(a,b) | a≥b} reflexive (x, x)
R
• {1, 2, 3, 4} reflexive
?
•R = {(1, 1), (2, 2), (2, 3), (3, 3), (4, 4)} Yes
•R = {(1, 1), (2, 2), (3, 3)} No
• : A
irreflexive (a, a) R a A
: < irreflexive9/5/2013

11. 11
Symmetry & Antisymmetry
• R A
symmetric
R = R−1, (a,b) R ↔ (b,a) R
– , = ( ) symmetric x = y y
= x x y <
symmetric
– “ ” symmetric, “ ”
• R
antisymmetric
a≠b, (a,b) R → (b,a) R
– : < antisymmetric, “ ”
• R A
asymmetric a,b A, (a,b) R9/5/2013

12. 12
• {1, 2, 3, 4}
symmetric, antisymmetric, asymmetric?
•R = {(1, 1), (1, 2), (2, 1), (3, 3), (4, 4)} symmetric
•R = {(1, 1)} sym. and
antisym.
•R = {(1, 3), (3, 2), (2, 1)} antisym. and
asym.
•R = {(4, 4), (3, 3), (1, 4)} antisym.
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13. 13
: R transitive
( a,b,c) (a,b) R (b,c) R → (a,c) R
• intransitive
:
– “ ” transitive
– “ ” intransitive
– = < > transitive
• {1, 2, 3, 4} transitive ?
•R = {(1, 1), (1, 2), (2, 2), (2, 1), (3, 3)} Yes
•R = {(1, 3), (3, 2), (2, 1)} No
•R = {(2, 4), (4, 3), (2, 3), (4, 1)} No
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14. 14
Exercise
• A={1,2,3,4} R,S,T,U
A
• R={(1,3), (3,4)}
• S={(1,1), (2,2), (3,3), (2,3), (3,2), (1,4)}
• T={(1,1), (2,2), (1,2), (2,1), (3,3), (4,4)}
• U={(1,1), (2,2), (3,3), (2,3), (3,2), (1,4),
(4,2), (1,2),(1,3)}
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15. 15
Answer
• R Irreflexive, Antisymmetric
Asymmetric
• S Transitive
• T Reflexive, Symmetric,
Transitive
• U
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16. 16
Composite
Relations
• R:A B, S:B C
SR R S :
SR = {(a,c) | b: aRb bSc}
• Rn R A
R0 :≡ IA ; Rn+1 :≡ RRn n≥0
IA = {(x,x)| x A}
R−n :≡ (R−1)n
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17. 17
Composite
Relations
: D S A = {1, 2, 3,
4}
D = {(a, b) | b = 5 - a} “b (5 – a)”
S = {(a, b) | a < b} “a b”
D = {(1, 4), (2, 3), (3, 2), (4, 1)}
S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
S D =
Q: R N : xRy y =
x2
S N : xSy y = x 3
S  R ?
A:
S  R xSRy y = x 6 (
{(2, 4), (3, 3), (3, 4), (4, 2), (4, 3), (4, 4)}
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18. 18
Q: R1 R2
R2  R1 :
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4
5 5R1 R2
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19. 19
1 1 1
2 2 2
3 3 3
4 4
5
A:
1:
1 1
2 2
3 3
4
R1 R2
R2  R19/5/2013

20. 20
1 1 1
2 2 2
3 3 3
4 4
5
A:
1:
1 1
2 2
3 3
4 R2  R1
R1 R2
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21. 21
1 1 1
2 2 2
3 3 3
4 4
5
A:
1:
1 1
2 2
3 3
4 R2  R1
R1 R2
9/5/2013

22. 22
1 1 1
2 2 2
3 3 3
4 4
5
A:
1:
1 1
2 2
3 3
4 R2  R1
R1 R2
9/5/2013

23. 23
1 1 1
2 2 2
3 3 3
4 4
5
A:
1:
1 1
2 2
3 3
4 R2  R1
R1 R2
R2  R1=
{(4,1),(4,2),(4,3)}
9/5/2013

24. 24
• : R A
Rn R n
• : R A
(a,b) R (b, c) R, (a, c) R a, b,
c A
• R
(a, c) R
R R R
R, R R R
• R R R
R (R R) R R,9/5/2013

25. 25
Representing
Relations)
• :
- Zero-one matrices
Directed graphs
• R
A = {a1, a2, …, am} B ={b1, b2, …, bn}, R
MR = [mij]
mij = 1, (ai, bj) R,
mij = 0, (ai, bj) R
•
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26. 26
• : R
{1, 2, 3} {1, 2} R
R = {(2, 1), (3, 1), (3, 2)} ?
• : MR
11
01
00
RM
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27. 27
• (
A A)
square matrices
•
Mref
1
.
.
.
1
1
Mref
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28. 28
•
MR = (MR)t
1101
1001
0010
1101
RM
0011
0011
0011
0011
RM
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29. 29
• -
0
0
101
0
0
01
1
0
0
0
0
1
1
1
1
Reflexive:
all 1‟s on diagonal
Irreflexive:
all 0‟s on diagonal
Symmetric:
all identical
across diagonal
Antisymmetric:
all 1‟s are across
from 0‟s
any-
thing
any-
thing
any-
thing
any-
thing
9/5/2013

30. 30
• : R S
MR MS
011
111
101
SRSR MMM
001
110
101
SM
• R S R S?
:
000
000
101
SRSR MMM
010
001
101
RM
9/5/2013

31. 31
• A = [aij] - m k
B = [bij] - k n
• Boolean product A B,
AB m n
i j [cij]
cij = (ai1 b1j) (ai2 b2j) … (aik bkj)
cij = 1 (ain bnj) = 1 n
(ain bnj) cij
= 09/5/2013

32. 32
• - MA = [aij], MB = [bij] MC = [cij]
A, B, C,
• MC = MAMB
cij = 1 (ain bnj) = 1
n
(ain bnj) cij = 0
C (xi, zj) yn
(xi, yn) A (yn, zj)
B
• C = B A (C A9/5/2013

33. 33
• B A
:
MB A = MAMB
•
A B
A Boolean product
B
•
:
[n]
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34. 34
• : R2
R
001
110
010
RM
: R2
010
111
110
]2[
2 RR
MM
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35. 35
Directed
Graphs
• directed graph) digraph)
G=(VG,EG) VG EG VG VG
R:A B
GR=(VG=A B, EG=R)
100
010
011
Mark
Fred
Joe
SallyMarySusan
R MR: R
GR: Joe
Fred
Mark
Susan
Mary
Sally
VG
EG
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36. 36
Directed
Graphs
• : V = {a, b, c, d},
E = {(a, b), (a, d), (b, b), (b, d), (c, a), (c, b), (d, b)}
a
b
cd
• (b, b) loop
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37. 37
Reflexive,
Symmetric
Directed Graphs






Reflexive: Irreflexive: Symmetric:
 
Antisymmetric:
 

asymmetric
antisymmetric
reflexive
irreflexive
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38. 38
(Equivalence
Relations)
•
• : A
(Reflexive)
(Symmetric) (Transitive)
•
R
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39. 39
(Equivalence Relations)
• : R
aRb l(a) = l(b),
l(x) x
R ?
• :
• R l(a) = l(a)
aRa a
• R l(a) = l(b)
l(b) = l(a)
aRb bRa
• R l(a) = l(b)
l(b) = l(c),
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40. 40
Equivalence Classes
• : R A
a
A a
• a R
[a]R [a]R :≡ { b | aRb }
•
a [a]
• b [a]R, b representative
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41. 41
Equivalence Classes
:
mouse,
[mouse] ?
: [mouse]
[mouse] ={horse, table, white,…}
„horse‟
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42. 42
Equivalence
Classes
• “ a b ”
– [a] =
a
• “ a b ”
– [a] = {a, −a}
• “ a b (
, a − b Z)”
– [a] = {…, a−2, a−1, a, a+1, a+2, …}
• “ a b
m ” ( m>1)
– [a] = {…, a−2m, a−m, a, a+m, a+2m, …}9/5/2013

43. 43
Equivalence Classes
: R A
:
• aRb
• [a] = [b]
• [a] [b]
: partition S
S
union
S Ai i I
S
(i) Ai i I
(ii) Ai Aj = , i j
(iii) A = S9/5/2013

44. 44
Partition
• : S {u, m, b, r, o, c, k,
s}
partition S
?
{{m, o, c, k}, {r, u, b, s}} yes
{{c, o, m, b}, {u, s}, {r}} no ( k)
{{b, r, o, c, k}, {m, u, s, t}} no (t S)
{{u, m, b, r, o, c, k, s}} yes
{{b, o, o, k}, {r, u, m}, {c, s}} yes ({b,o,o,k} = {b,o,k})
{{u, m, b}, {r, o, c, k, s}, } no ( )
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45. 45
Equivalence Classes
• : R S
R
partition S
{Ai | i I} S
R Ai, i I,
• R
S S
S
9/5/2013

46. 46
Equivalence Classes
• : ,
,
• R {(a, b) | a b
} P = {
}
R = {( , ), ( , ), ( , ), (
, ), ( , ), ( , ), ( ,
), ( , ), ( , ), ( , ),
( , ), ( , ), ( , ), ( ,
)}
• R :9/5/2013

47. 47
Equivalence Classes
• : R
{(a, b) | a b (mod 3)}
R (a, b) | a
b }
• R ?
• R
• R ?
{{…, -6, -3, 0, 3, 6, …},
{…, -5, -2, 1, 4, 7, …},
{…, -4, -1, 2, 5, 8, …}}9/5/2013