Relations (1)

1
Relations (1)

• Rosen 6th ed., ch. 8

2
Binary Relations

• Let A, B be any two sets.
• A binary relation R from A to B, written (with
  signature) RA?B, is a subset of AB.
• E.g., let lt N?N (n,m) n lt m
• The notation a R b or aRb means (a,b)?R.
• E.g., a lt b means (a,b)? lt
• 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 PRAB?T,F defined over the 2 sets
  A,B e.g., eats (a,b) organism a eats
  food b

3
Complementary Relations

• Let RA?B be any binary relation.
• Then, RA?B, the complement of R, is the binary
  relation defined by R (a,b) (a,b)?R
  (AB) - R
• Note this is just R if the universe of discourse
  is U AB thus the name complement.
• Note the complement of R is R.

Example lt (a,b) (a,b)?lt (a,b) altb

4
Inverse Relations

• Any binary relation RA?B has an inverse relation
  R-1B?A, defined by R-1 (b,a) (a,b)?R.
• E.g., lt-1 (b,a) altb (b,a) bgta gt.
• E.g., if RPeople?Foods is defined by
  aRb ? a eats b, then b R-1 a ? b is eaten
  by a. (Passive voice.)

5
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 ? 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)

6
Reflexivity

• A relation R on A is reflexive if ?a?A, aRa.
• E.g., the relation (a,b) ab is
  reflexive.
• A relation is irreflexive iff its complementary
  relation is reflexive. (for every a?A, (a, a) ?
  R)
• Note irreflexive ? not reflexive!
• Example lt is irreflexive.
• Note likes between people is not reflexive,
  but not irreflexive either. (Not everyone likes
  themselves, but not everyone dislikes themselves
  either.)

7
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. lt 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))
• lt is antisymmetric, likes is not.

8
Asymmetry

• A relation R is called asymmetric if (a,b)?R
  implies that (b,a) ? R.
• is antisymmetric, but not asymmetric.
• lt is antisymmetric and asymmetric.
• likes is not antisymmetric and asymmetric.

9
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 ?

10
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).

11
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

12
Examples Cont.

• Example 5. Relations on the set of integers
• R1 (a, b) a b, R2 (a, b) a gt 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,

13
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

14
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.

15
Composition of Relations Cont.

• R relation between A and B
• S relation between B and C
• SR composition of relations R and S
• A relation between A and C
• (x,z) x Î A, z Î C, and
• there exists y Î B such that xRy and ySz

B
C
C
A
S
A
R
z
SR
z
y
x
x
u
w
w
16
Example
R
S
1 2 3 4
1 2 3 4
1 2 3
1 2
SR
1 2 3
1 2
17
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)

18
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 Rn1 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) gt Rn R3

19
Power of Relations Cont.

• THEOREM The relation R on a set A is transitive
  if and only if Rn Í R for n 1, 2, 3, .
• EXAMPLE Let R (1, 1), (2, 1), (3, 2), (4,
  3). Is R transitive? gt 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 ? gt maybe a tedious
  task
  

20
Representing Relations

• Some special ways to represent binary relations
• With a zero-one matrix.
• With a directed graph.

21
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 representationof that
  Likesrelation

22
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.

any-thing
any-thing
anything
anything
any-thing
any-thing
Symmetricall identicalacross diagonal
Antisymmetricall 1s are acrossfrom 0s
Reflexiveall 1s on diagonal
Irreflexiveall 0s on diagonal
23
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
• ?
  

24
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).

25
Examples Cont.

• Determine properties of these relations on 1, 2,
  3, 4.
• R1
  R2 R3

26
Examples Cont.

• Solution
• Reflective X, O, X
• Irreflective O, X, O
• Symmetric O, X, O
• Antisymmetric X, X, X
• Asymmetric X, X, X

27
Examples Cont.

• Transitive
• R12 ?
  
• Not Transitive
• Notice (1, 4) and (4, 3) are in R1 but not (1,
  3)

28
Examples Cont.

• R22 ?
  
• R23 ?
  

29
Examples Cont.

• R24 ?
  
• Not transitive
• Notice (1, 3) and (3, 4) are in R1 but not (1,
  4)

30
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 ?
  
• R3 ?
  
• R33 ?
  

31
Using Directed Graphs

• A directed graph or digraph G(VG,EG) is a set VG
  of vertices (nodes) with a set EG?VGVG of edges
  (arcs,links). Visually represented using dots
  for nodes, and arrows for edges. Notice that a
  relation RA?B can be represented as a graph
  GR(VGA?B, EGR).

Edge set EG(blue arrows)
GR
MR
Joe
Susan
Fred
Mary
Mark
Sally
Node set VG(black dots)
32
Digraph Reflexive, Symmetric

• It is extremely easy to recognize the
  reflexive/irreflexive/ symmetric/antisymmetric
  properties by graph inspection.

?
?
?
?
?
?
?
?
?
?
?
ReflexiveEvery nodehas a self-loop
IrreflexiveNo nodelinks to itself
SymmetricEvery link isbidirectional
AntisymmetricNo link isbidirectional
Asymmetric, non-antisymmetric
Non-reflexive, non-irreflexive
33
Example

• Determine which are reflexive, irreflexive,
  symmetric, antisymmetric, and transitive.

transitive
Reflexive Symmetric Not transitive
Reflexive antisymmetric