Relations (1)
Rosen 6th ed., ch. 8
Binary Relations
• Let A, B be any two sets.
• A binary relation R from A to B, written (with signature)
R:A↔B, is a subset of A×B.
– E.g., let < : N↔N :≡ {(n,m) | n < m}
• The notation a R b or aRb means (a,b)R.
– E.g., a < b means (a,b) <
• If aRb we may say “a is related to b (by relation R)”, or
“a relates to b (under relation R)”.
• A binary relation R corresponds to a predicate function
PR:A×B→{T,F} defined over the 2 sets A,B; e.g.,
“eats” :≡ {(a,b)| organism a eats food b}
Complementary Relations
• Let R:A↔B be any binary relation.
• Then, R:A↔B, the complement of R, is the
binary relation defined by
R :≡ {(a,b) | (a,b)R} = (A×B) − R
Note this is just R if the universe of discourse
is U = A×B; thus the name complement.
• Note
the< complement
of=R{(a,b)
is R.| ¬a<b} = ≥
Example:
= {(a,b) | (a,b)<}
Inverse Relations
• Any binary relation R:A↔B has an inverse
relation R−1:B↔A, defined by
R−1 :≡ {(b,a) | (a,b)R}.
E.g., <−1 = {(b,a) | a<b} = {(b,a) | b>a} = >.
• E.g., if R:People→Foods is defined by
aRb  a eats b, then:
b R−1 a  b is eaten by a. (Passive voice.)
Relations on a Set
• A relation on the set A is a relation from A to
A.
• In other words, a relation on a set A is a
subset of A х A.
• Example 4. Let A be the set {1, 2, 3, 4}. Which
ordered pairs are in the relation R = {(a, b) | a
divides b}?
Solution: R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2),
(2, 4), (3, 3), (4, 4)}
Reflexivity
• A relation R on A is reflexive if aA, aRa.
– E.g., the relation ≥ :≡ {(a,b) | a≥b} is reflexive.
• A relation is irreflexive iff its complementary relation
is reflexive. (for every aA, (a, a)  R)
– Note “irreflexive” ≠ “not reflexive”!
– Example: < is irreflexive.
– Note: “likes” between people is not reflexive, but not
irreflexive either. (Not everyone likes themselves, but not
everyone dislikes themselves either.)
Symmetry & Antisymmetry
• A binary relation R on A is symmetric iff R = R−1,
that is, if (a,b)R ↔ (b,a)R.
i.e, ab((a, b)  R → (b, a)  R))
– E.g., = (equality) is symmetric. < is not.
– “is married to” is symmetric, “likes” is not.
• A binary relation R is antisymmetric if
ab(((a, b)  R  (b, a)  R) → (a = b))
– < is antisymmetric, “likes” is not.
Asymmetry
• A relation R is called asymmetric if (a,b)R
implies that (b,a)  R.
– ‘=’ is antisymmetric, but not asymmetric.
– < is antisymmetric and asymmetric.
– “likes” is not antisymmetric and asymmetric.
Transitivity
• A relation R is transitive iff (for all a,b,c)
(a,b)R  (b,c)R → (a,c)R.
• A relation is intransitive if it is not transitive.
• Examples: “is an ancestor of” is transitive.
• “likes” is intransitive.
• “is within 1 mile of” is… ?
Examples of Properties of Relations
• Example 7. Relations on {1, 2, 3, 4}
– R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)},
R2 = {(1, 1), (1, 2), (2, 1)},
R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1),
(4, 4)},
R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)},
R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4),
(3, 3), (3, 4), (4, 4)},
R6 = {(3, 4)}.
Examples Cont.
• Solution
– Reflective : X, X, O, X, O, X
– Irreflective : X, X, X, O, X, O
– Symmetric : X, O, O, X, X, X
– Antisymmetric : X, X, X, O, O, O
– Asymmetric : X, X, X, O, X, O
– Transitive : X, X, X, O, O, O
Examples Cont.
• Example 5. Relations on the set of integers
– R1 = {(a, b) | a ≤ b},
R2 = {(a, b) | a > b},
R3 = {(a, b) | a = b or a = -b},
R4 = {(a, b) | a = b},
R5 = {(a, b) | a = b + 1},
R6 = {(a, b) | a + b ≤ 3},
Examples Cont.
• Solution
– Reflective : O, X, O, O, X, X
– Irreflective : X, O, X, X, O, X
– Symmetric : X, X, O, O, X, O
– Antisymmetric : O, O, X, O, O, X
– Asymmetric : X, O, X, X, O, X
– Transitive : O, O, O, O, X, X
Composition of Relations
• DEFINITION: Let R be a relation from a set A to
a set B and S 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.
Composition of Relations Cont.
• R : relation between A and B
• S : relation between B and C
• S°R : composition of relations R and S
– A relation between A and C
– {(x,z)| x  A, z  C, and
there exists y  B such that xRy and ySz }
B
A
R
x
S
y
u
C
z
w
C
A
S°R
z
x
w
Example
R
1
2
3
S
1
1
2
2
1
3
3
2
4
4
S°R
1
1
2
2
3
Example Cont.
• What is the composite of the relations R and S,
where R is the relation from {1, 2, 3} to {1, 2, 3,
4} with R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)}
and S is the relation from {1, 2, 3, 4} to {0, 1, 2}
with S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}.
Solution:
S◦R = {(1, 0), (1, 1), (2, 1), (2, 2), (3, 0), (3, 1)}
Power of Relations
• DEFINITION: Let R be a relation on the set A.
The powers Rn, n = 1, 2, 3, …, are defined
recursively by R1 = R and Rn+1 = Rn◦R.
• EXAMPLE: Let R = {(1, 1), (2, 1), (3, 2), (4, 3)}.
Find the powers Rn, n = 2, 3, 4, ….
• Solution:
R2 = R◦R = {(1, 1), (2, 1), (3, 1), (4, 2)},
R3 = R2◦R = {(1, 1), (2, 1), (3, 1), (4, 1)},
R4 = R2◦R = {(1, 1), (2, 1), (3, 1), (4, 1)} => Rn = R3
Power of Relations Cont.
• THEOREM: The relation R on a set A is
transitive if and only if Rn  R for n = 1, 2, 3, ….
• EXAMPLE: Let R = {(1, 1), (2, 1), (3, 2), (4, 3)}.
Is R transitive? => No (see the previous page).
• EXAMPLE: Let R = {(2, 1), (3, 1), (3, 2), (4, 1), (4,
2), (4, 3)}. Is R transitive?
• Solution:
R2 = ?, R3 = ?, R4 = ? => maybe a tedious task^^
Representing Relations
• Some special ways to represent binary
relations:
– With a zero-one matrix.
– With a directed graph.
Using Zero-One Matrices
• To represent a relation R by a matrix
MR = [mij], let mij = 1 if (ai,bj)R, else 0.
• E.g., Joe likes Susan and Mary, Fred likes Mary,
and Mark likes Sally.
• The 0-1 matrix
Susan Mary Sally
representation
Joe  1
1
0 
of that “Likes”


Fred  0
1
0 
relation:
Mark  0
0
1 
Zero-One Reflexive, Symmetric
• Terms: Reflexive, non-Reflexive, irreflexive,
symmetric, asymmetric, and antisymmetric.
– These relation characteristics are very easy to
recognize by inspection of the zero-one matrix.
1

any 1 thing



 any- 1 
 thing

1

Reflexive:
all 1’s on diagonal
0

any 0
thing 


 any- 0 
 thing

0

Irreflexive:
all 0’s on diagonal
 1

1

0






0
 Symmetric:

all identical
across diagonal
 0

1

0
1


 0



0
 Antisymmetric:

all 1’s are across
from 0’s
Finding Composite Matrix
• Let the zero-one matrices for S◦R, R, S be M S◦R,
MR, MS. Then we can find the matrix
representing the relation S◦R by:
•
M S◦R = MR ⊙ MS
• EXAMPLE: MR
MS
M S◦R
0 1 0 
1 0 1
0 1 0 
0 1 0 


⊙
0 1 0
1 0 1
0 1 0 
0 1 0 
=

0 1 0
Examples of matrix representation
• List the ordered pairs in the relation on {1, 2, 3}
corresponding to these matrices (where the
rows and columns correspond to the integers
listed in increasing order).
1 0 1
0 1 0 


1 0 1
0 1 0 
0 1 0 


0 1 0
1 1 1
1 0 1


1 1 1
Examples Cont.
• Determine properties of these relations on {1,
2, 3, 4}.
R1
1
1

0

1
1 0 1
0 1 0
1 1 1

0 1 1
R2
1
0

0

1
1 1 0
1 0 0
0 1 1

0 0 1
R3
0
1

0

1
1 0 1
0 1 0
1 0 1

0 1 0
Examples Cont.
• Solution
– Reflective : X, O, X
– Irreflective : O, X, O
– Symmetric : O, X, O
– Antisymmetric : X, X, X
– Asymmetric : X, X, X
Examples Cont.
– Transitive :
1
1
– R12 = 
0

1
1 0 1
1
0 1 0 ⊙ 1

0
1 1 1


0 1 1
1
1 0 1 1
0 1 0 = 1
1 1 1 1
 
0 1 1 1
1 1 1
1 1 1
1 1 1

1 1 1
– Not Transitive
– Notice: (1, 4) and (4, 3) are in R1 but not (1, 3)
Examples Cont.
1
0
– R22 = 0

1
1 1 0
1
0
1 0 0

⊙
0
0 1 1


0 0 1
1
1 1 0  1
1 0 0 0
=
0 1 1  1
 
0 0 1  1
1 1 1
1 0 0
0 1 1

1 1 1
1
0

3
– R 2 = 1

1
1 1 1
1
0
1 0 0

0 1 1  ⊙ 0


1 1 1
1
1 1 0 1
1 0 0 0
0 1 1 = 1
 
0 0 1 1
1 1 1
1 0 0
1 1 1

1 1 1
Examples Cont.
1
0
4
– R2 = 
1

1
1 1 1
1
1 0 0 ⊙ 0
0
1 1 1


1 1 1
1
1 1 0 1
1 0 0 = 0
0 1 1 1
 
0 0 1 1
1 1 1
1 0 0
1 1 1

1 1 1
– Not transitive
– Notice: (1, 3) and (3, 4) are in R1 but not (1, 4)
Examples Cont.
• EXAMPLE: Let R = {(2, 1), (3, 1), (3, 2), (4, 1), (4,
2), (4, 3)}. Is R transitive?
• Solution: transitive (see below)
– R2
0
1

1

1
0 0 0
0 0 0
1 0 0

1 1 0
– R3
0
0

1

1
0 0 0
0 0 0
0 0 0

1 0 0
⊙
0
1

1

1
0 0 0
0 0 0
1 0 0

1 1 0
⊙
0
1

1

1
0 0 0
0 0 0
1 0 0

1 1 0
=
0
0

1

1
0 0 0
0 0 0
0 0 0

1 0 0
=
0
0

0

1
0 0 0
0 0 0
0 0 0

0 0 0
Using Directed Graphs
• A directed graph or digraph G=(VG,EG) is a set VG of
vertices (nodes) with a set EGVG×VG of edges (arcs,links).
Visually represented using dots for nodes, and arrows for
edges. Notice that a relation R:A↔B can be represented
as a graph GR=(VG=AB, EG=R).
Susan Mary Sally
MR
Joe  1
1
0 


Fred  0
1
0 
Mark  0
0
1 
GR
Edge set EG
(blue arrows)
Joe
Fred
Susan
Mary
Mark
Sally
Node set VG
(black dots)
Digraph Reflexive, Symmetric
It is extremely easy to recognize the reflexive/irreflexive/
symmetric/antisymmetric properties by graph inspection.
 



 
Reflexive:
Every node
has a self-loop
 
Irreflexive:
No node
links to itself
Asymmetric, non-antisymmetric
Symmetric:
Every link is
bidirectional
 
Antisymmetric:
No link is
bidirectional
Non-reflexive, non-irreflexive
Example
• Determine which are reflexive, irreflexive,
symmetric, antisymmetric, and transitive.
b
a
a
transitive
c
a
c
b
c
Reflexive
antisymmetric
b
Reflexive
Symmetric
Not transitive