Chapter 7 Relations : the second time around

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Chapter 7 Relations the second time around

• Yen-Liang Chen
• Dept of Information Management
• National Central University

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7.1. Relations revisited properties of relations

• Definition 7.1. For sets A, B, any subset of A?B
  is called a (binary) relation from A to B. Any
  subset of A?A is called a binary relation on A .

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Examples

• Ex 7.1
• Define the relation ? on the set Z by a?b, if
  a?b.
• For x, y?Z and n?Z, the modulo n relation ? is
  defined by x?y if x-y is a multiple of n.
• Ex 7.2 For x, y ?A, define x?y if x is a prefix
  of y.

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Ex 7.3.

• s1?s2 if v(s1,x)s2. Here, ? denotes the first
  level of reachability.
• s1?s2 if v(s1,x1x2)s2. Here, ? denotes the
  second level of reachability.
• 1-equivalence relation s1E1s2 if w(s1,x)w(s2,x)
  for x?IA.
• k-equivalence relation s1Eks2 if w(s1,x)w(s2,x)
  for x?IAk.
• equivalence relation if two states are
  k-equivalent for all k.

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reflexive

• Definition 7.2. A relation ? on a set A is called
  reflexive if for all x?A, (x, x)??.
• Ex 7.4. For A1, 2, 3, 4, a relation ??A?A will
  be reflexive if and only if ??(1, 1), (2, 2),
  (3, 3), (4, 4).
• Ex 7.5. Given a finite set A with ?A?n, we have
  ?A?A?n2, so there are relations on A. Among
  them, is reflexive.

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symmetric

• Definition 7.3. A relation ? on a set A is called
  symmetric if (x, y)??? (y, x)?? for all x, y?A.
• Ex 7.6. With A1, 2, 3, what properties do the
  following relations have?
• ?1(1, 2), (2, 1), (1, 3), (3, 1)
• ?2(1, 1), (2, 2), (3, 3), (2, 3)
• ?3(1, 1), (2, 2), (3, 3)
• ?4(1, 1), (2, 2), (2, 3), (2, 3), (3, 2)
• ?5(1, 1), (2, 3), (3, 3)

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symmetric

• To count the symmetric relations on Aa1, a2,,
  an.
• A?AA1?A2, where A1(a1, a1),, (an, an) and
  A2(ai, aj)?i?j.
• A1 contains n pairs, and A2 contains n2-n pairs.
• A2 contains (n2-n)/2 subsets Si,j of the form
  (ai, aj), (aj, ai)?i?j.
• So, we have totally
  symmetric relations on A.
• If the relations are both symmetric and
  reflexive, we have choices.

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transitive

• Definition 7.4. A relation ? on a set A is called
  transitive if (x,y), (y,z)??? (x,z)?? for all x,
  y, z?A.
• Ex 7.8. Define the relation ? on the set Z by
  a?b if a divides b. This is a transitive and
  reflexive relation but not symmetric.
• Ex 7.9. Define the relation ? on the set Z by a?b
  if a?b?0. What properties do they have?

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anti-symmetric

• Definition 7.5. A relation ? on a set A is called
  anti-symmetric if (x, y)?? and (x, y)?? ? xy for
  all x, y?A.
• Ex 7.11. Define the relation (A, B)?? if A?B.
  Then it is an anti-symmetric relation.
• Note that not symmetric is different from
  anti-symmetric.
• Ex 7.12. What properties do they have?
• ?(1, 2), (2, 1), (2, 3)
• ?(1, 2), (2, 2)

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anti-symmetric

• To count the anti-symmetric relations on Aa1,
  a2,, an.
• A?AA1?A2, where A1(a1, a1),, (an, an) and
  A2(ai, aj)?i?j.
• A1 contains n pairs, and A2 contains n2-n pairs.
• A2 contains (n2-n)/2 subsets Si,j of the form
  (ai, aj), (aj, ai)?i?j.
• Each element in A1 can be selected or not.
• Each element in Si,j can be selected either one
  or none.
• So, we have totally
  anti-symmetric relations on A.
• Ex 7.13. Define the relation ? on the functions
  by f ? g if f is dominated by g (or f?O(g)). What
  are their properties?

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partial order relation

• Definition 7.6. A relation ? is called a partial
  order, if ? is reflexive, anti-symmetric and
  transitive.
• Ex 7.15. Define the relation ? on the set Z by
  a?b if a divides b.

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equivalence relation

• Definition 7.7. A relation ? is called an
  equivalence relation, if ? is reflexive,
  symmetric and transitive.
• Ex 7.16.(b)
• If A1, 2, 3, the following are all equivalence
  relations
• ?1(1, 1), (2, 2), (3, 3)
• ?2(1, 1), (2, 2), (3, 3), (2, 3), (3,2)
• ?3(1, 1), (1, 3), (2, 2), (3, 1), (3, 3)
• ?4(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2,
  3), (3, 1), (3, 2), (3, 3)

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Examples

• Ex 7. 16(c). For a finite set A, A ?A is the
  largest equivalence relation on A. The equality
  relation is the smallest equivalence relation on
  A.
• Ex 7.16(d). Let f A?B be the onto function.
  Define the relation ? on A by a?b if f(a)f(b).
• Ex 7. 16(e). If ? is a relation on A, then ? is
  both an equivalence relation and a partial order
  relation iff ? is the equality relation on A.

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7.2. Computer recognition zero-one matrices and
directed graphs

• Definition 7.8. Let relations ?1?A?B and ?2?B?C.
  The composite relation of ?1??2 is a relation
  defined by ?1??2(x, z)? ?y in B such that (x,
  y)? ?1 and (y, z)? ?2.
• Ex 7.17. Consider ?1(1, x), (2, x), (3, y), (3,
  z) and ?2(w, 5), (x, 6). What is ?1??2?

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composite relation

• Ex 7. 18. Let A be the set of employees at a
  computer center, while B denotes a set of
  programming language, and C is a set of
  projects
• Theorem 7.1.
• ?1?(?2??3) (?1??2)??3

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the power of relation

• Definition 7.9. We define the powers of relation
  ? by (a) ?1? (b) ?n1?? ?n
• Ex 7.19. If ?(1, 2), (1, 3), (2, 4), (3, 2),
  then what is ?2 and ?3 and ?4.

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matrix representation

• A relation can be represented by an m?n zero-one
  matrix.
• Ex 7.17. Consider ?1(1, x), (2, x), (3, y), (3,
  z) and ?2(w, 5), (x, 6). What is ?1??2?

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matrix representation

• Ex 7.19. If ?(1, 2), (1, 3), (2, 4), (3, 2),
  then what is ?2 and ?3 and ?4.

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matrix representation

• Let A be a set with ?A?n and ? be a relation on
  A. If M(?) is the relation matrix for ?, then
• M(?)0 if and only if ??.
• M(?)1 if and only if ?A?A.
• M(?m)M(?)m

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less than

• Definition 7.11. Let E(eij)m?n F(fij)m?n be two
  zero-one matrices. We say that E precedes, or is
  less than , F, written as E?F, if eij? fij for
  all i, j.
• Ex 7.23. E?F.

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Identity matrix
I2
I3
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Transpose of a matrix
A
Atr
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Theorem 7.2

• Let M denote the relation matrix for ?. Then
• (A) R is reflexive if and only if In?M.
• (B) R is symmetric if and only if MMtr.
• (C) R is transitive if and only if M2?M.
• (D) R is anti-symmetric if and only if M?Mtr?In.

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Graph representation

• Definition 7.14. A directed graph can be denoted
  as G(V, E), where V is the vertex set and E is
  the edge set.
• V1,2,3,4,5, E(1,1),(1,2),(1,4),(3,2)

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Ex 7.27

• R(1,1),(1,2),(2,3),(3,2),(3,3),(3,4),(4,2)
• directed graph, undirected graph, connected,
  undirected cycle, directed cycle

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Terms in graph

• strongly connected and loop-free
• disconnected graph, components

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complete graphs
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Graph representation for a relation

• Ex 7.30, Fig 7.8, ? is reflexive if and only if
  its directed graph contains a loop at each vertex
• Ex 7.31, Fig 7.9, ? is symmetric if and only if
  its directed graph may be drawn only by loops and
  undirected edges
• Ex 7.32, Fig 7.10, ? is anti-symmetric if and
  only if for any x?y the graph contains at most
  one of the edges (x, y) or (y, x)
• Ex 7.33, Fig 7.11, a relation is an equivalence
  relation if and only if its graph consists of
  disjoint union of complete graphs augmented by
  loops at each vertex

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7.3. Partial orders Hasse Diagrams

• Definition Let A be a set with ? a relation on
  A. The pair (A, ?) is called a partially ordered
  set, or poset, if relation ? on A is partially
  ordered. If A is called a poset, we understand
  that there is a partially order ? on A that makes
  A into this set.

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Examples of Poset

• Ex 7.34. Let A be the set of courses offered at
  a college. Define the relation ? on A by x?y if x
  ,y are the same course or if x is a prerequisite
  for y.
• Ex 7.35. Define ? on A1, 2, 3, 4 by x?y if x
  divide y. Then (A, ?) is a poset.
• Ex 7.36. Let A be the set of tasks that must be
  performed to build a house. Define the relation ?
  on A by x?y if x ,y are the same task or if x
  must be performed before y.

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Original graph and Hasse diagram
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Hasse Diagram

• If (A, ?) is a poset, we construct a Hasse
  diagram for ? on A by drawing a line segment from
  x up to y, if
• x?y
• there is no z such that x?z and z?y.
• Ex 7.38, Fig 7.18. The relation on (a) is the
  subset relation, while the relations on the
  others are the divide relations.

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totally ordered

• Definition 7.16. If (A, ?) is a poset, we say
  that A is totally ordered if for all x, y ?A
  either x?y or y?x. In this case, ? is called a
  total order.
• Ex 7.40.
• On the set N, the relation ? defined by x?y if
  x?y is a total order.
• The subset relation is a partial order but not
  total order.
• Fig 7.19 is a total order.

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Topological sorting

• Given a Hasse diagram for a partial order
  relation ?, how to find a total order ? for which
  ???.

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maximal and minimal

• Definition 7.17. If (A, ?) is a poset, then x is
  a maximal element of A if for all a?A, a?x?x a.
  Similarly, y is a minimal element of A if for all
  b?A, b?y?b y .
• Ex 7.42.
• For the poset (P(U), ?), U is the maximal and ?
  is the minimal.
• Let B be the proper subsets of 1, 2, 3. Then we
  have multiple maximal elements for the poset (B,
  ?).

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Examples

• Ex 7.43. For the poset (Z, ?), we have neither a
  maximal nor a minimal element. For the poset (N,
  ?), we have no maximal element but a minimal
  element 0.
• Ex 7.44. How about the poset in Fig. 7.18? Do
  they have maximal or minimal elements?
• Theorem 7.3. If (A, ?) is a poset and A is
  finite, then A has both a maximal and a minimal
  element.

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Least and greatest

• Definition 7.18. If (A, ?) is a poset, then x is
  a least element of A if for all a?A, x?a.
  Similarly, y is a greatest element of A if for
  all a?A, a?y.
• Ex 7.45.
• For the poset (P(U), ?), U is the greatest and ?
  is the least.
• Let B be the nonempty subsets of 1, 2, 3. Then
  we have U as the greatest maximal element and
  three minimal elements for the poset (B, ?).
• Theorem 7.4. If poset (A, ?) has a greatest or a
  least element, then that element is unique.

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Lower bound and upper bound

• Definition 7.19. If (A, ?) is a poset with B?A,
  then x?A is called a lower bound of B if x?b for
  all b?B. Likewise, y?A is called an upper bound
  of B if b?y for all b?B.
• An element x??A is called a greatest lower bound
  of B if for all other lower bounds x? of B we
  have x??x?. Similarly, an element x??A is called
  a least upper bound of B if for all other upper
  bounds x? of B we have x??x?.
• Theorem 7.5. If (A, ?) is a poset and B?A, then B
  has at most one lub (glb).

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Examples

• Ex 7.47. Let U1, 2, 3, 4 with AP(U) and let
  ? be the subset relation on B. If B1, 2,
  1, 2, then what are the upper bounds of B,
  lower bounds of B, the greatest lower bound and
  the least upper bound?
• Ex 7.48. Let ? be the ? relation on A. What are
  the results for the following cases?
• AR and B0, 1
• AR and Bq?Q?q2lt2
• AQ and Bq?Q?q2lt2

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Lattice

• Definition 7.20. The poset (A, ?) is called a
  lattice if for all x, y?A the elements lubx, y
  and glbx, y both exist in A.
• Ex 7.49. For AN and x, y?N, define x?y by x?y.
  Then lubx, ymaxx, y and glbx, yminx, y.
  (N,?) is a lattice.
• Ex 7.50. For the poset (P(U), ?), if S, T?U, we
  have lubS, TS?T and glbS, TS?T and it is a
  lattice.

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7.4. Equivalence relation and partitions

• For any set A??, the relation of equality is an
  equivalence relation on A.
• Let the relation on Z defined by x?y if x-y is a
  multiple of 2, then ? is an equivalence relation
  on Z, where one contains all even integers and
  the other odd integers.

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partition

• Definition 7.21. Given a set A and index set I,
  let ??Ai?A for i?I. Then Aii?I is a partition
  of A if (a) A?i?IAi and (b) Ai?Aj? for i?j.
• Ex 7.52, A1,,10.
• Ex 7.53. A partition of R

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equivalence class

• Definition 7.22. the equivalence class of x,
  denoted x, is defined by xy?A?y?x
• Ex 7.54. Define the relation ? on Z by x?y if
  4?(x-y).
• Ex 7.55. Define the relation ? on Z by a?b if
  a2b2.

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equivalence class

• Theorem 7.6. If ? is an equivalence relation on a
  set A and x, y?A, then (a) x?x (b) x?y if and
  only if xy and (c) xy or x?y?.
• Ex 7.56.
• Let A1, 2, 3, 4, 5, ?(1, 1), (2, 2), (2, 3),
  (3, 2), (3, 3), (4, 4), (4, 5), (5, 4), (5, 5),
  Then, we have A1?2?4.
• Consider an onto function fA?B. f(1, 3, 7)x
  f(4, 6)y f(2, 5)z. The relation ? defined
  on A by a?b if f(a)f(b). A1?4?2.
• Ex 7.58. If an equivalence relation ? on A1,
  2, 3, 4, 5, 6, 7 induces the partition A1, 2?
  3?4, 5, 7?6, what is ??

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Theorems

• Theorem 7.7. If A is a set, then any equivalence
  relation ? on A induces a partition of A, and any
  partition of A gives rise to an equivalence
  relation ? on A.
• Theorem 7.8. For any set A, there is one-to-one
  correspondence between the set of equivalence
  relations on A and the set of partitions of A.

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7.5. Finite state machine the minimization
process

• Two finite state machines of the same function
  may have different number of internal states.
• Some of these states are redundant.
• A process of transforming a given machine into
  one that has no redundant internal states is
  called the minimization process.

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1-equivalence

• For the states S, we define the relation E1 on S
  by s1E1s2 if w(s1, x)w(s2, x) for all x?I. The
  relation E1 is an equivalence relation on S, and
  it partitions S into subsets such that two states
  are in the same subset if they produce the same
  output for each x?I.

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k-equivalence

• For the states S, we define the k-equivalence
  relation Ek on S by s1Eks2 if w(s1, x)W(s2, x)
  for all x?Ik. The relation Ek is an equivalence
  relation on S, and it partitions S into subsets
  such that two states are in the same subset if
  they produce the same output for each x?Ik.
• Finally, we call two states equivalent if they
  are k-equivalent for all k?1.

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Goal and tips

• Hence, our objective is to determine the
  partition of S induced by E and to select one
  state for each equivalent class.
• Observations
• If two states are not 2-equivalent, they can not
  be 3-equivalent.
• For s1, s2?S, where s1Eks2, we find that s1Ek1s2
  if and only if v(s1, x)Ekv(s2,x) for all x?I.

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The procedure for the minimization

• Set k1. We determine the states that are
  1-equivalent.
• Having determined Pk, we determine the states
  that are (k1)-equivalent. Note that if s1Eks2,
  then s1Ek1s2 if and only if v(s1, x)Ekv(s2,x)
  for all x?I.
• If Pk1Pk, the process is completed.

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Ex 7.60.

• the original table in Table 7.1 and P1s1, s2,
  s5, s6, s3, s4
• Table 7.2. P2s1, s2, s5, s6, s3, s4,
  P2P3

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refinement

• Definition 7.23. If P1 and P2 are partitions of
  set A, then P2 is called a refinement of P1,
  denoted as P2?P1, if every cell of P2 is
  contained in a cell of P1. When P2?P1 and P2?P1,
  we write P2ltP1.
• Theorem 7.9. In the minimization process, if
  Pk1Pk, then Pr1Pr for all r?k1.

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distinguishing string

• If s1Eks2 but s1?Ek1s2, then we have a string
  xx1x2xkxk1?Ik1 such that w(s1, x)?w(s2, x)
  but w(s1, x1x2xk)w(s2, x1x2xk). We call this
  string as distinguishing string.
• s1?Ek1s2??x1?I v(s1, x1) ?Ek v(s2, x1)

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Ex 7.61

• s2E1s6 but s2?E2s6.
• X00 is the minimal distinguishing string for s2
  and s6

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Ex 7.62

• s1 and s4 are 2 equivalent but are not
  3-equivalent.
• X111 is the minimal distinguishing string for s1
  and s4