1. Chapter 7 Relations : the second time around
Yen-Liang Chen
Dept of Information Management
National Central University

2. 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 .

3. Examples Ex 7.1 Ex 7.2: For x, y A, define xy if x is a prefix of y.
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.

4. 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.

5. reflexive
Definition 7.2. A relation  on a set A is called reflexive if for all xA, (x, x).
Ex 7.4. For A={1, 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.

6. 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 A={1, 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)}

7. symmetric To count the symmetric relations on A={a1, a2,…, an}.
AA=A1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.

8. 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?

9. anti-symmetric
Definition 7.5. A relation  on a set A is called anti-symmetric if (x, y) and (x, y)  x=y for all x, yA.
Ex 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 What properties do they have?
={(1, 2), (2, 1), (2, 3)}
={(1, 2), (2, 2)}

10. anti-symmetric
To count the anti-symmetric relations on A={a1, a2,…, an}.
AA=A1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 Define the relation  on the functions by f  g if f is dominated by g (or fO(g)). What are their properties?

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

12. equivalence relation
Definition 7.7. A relation  is called an equivalence relation, if  is reflexive, symmetric and transitive.
Ex 7.16.(b)
If A={1, 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)}

13. 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.

14. 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 Consider 1={(1, x), (2, x), (3, y), (3, z)} and 2={(w, 5), (x, 6)}. What is 12?

15. composite relation
Ex 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

16. the power of relation
Definition 7.9. We define the powers of relation  by (a) 1=; (b) n+1= n
Ex If ={(1, 2), (1, 3), (2, 4), (3, 2)}, then what is 2 and 3 and 4.

17. matrix representation
A relation can be represented by an mn zero-one matrix.
Ex Consider 1={(1, x), (2, x), (3, y), (3, z)} and 2={(w, 5), (x, 6)}. What is 12?

18. matrix representation
Ex If ={(1, 2), (1, 3), (2, 4), (3, 2)}, then what is 2 and 3 and 4.

19. 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

20. less than
Definition 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 EF.

21. Identity matrix
I2=
I3=

22. Transpose of a matrix A=
Atr=

23. 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 M=Mtr.
(C) R is transitive if and only if M2M.
(D) R is anti-symmetric if and only if MMtrIn.

24. Graph representation
Definition A directed graph can be denoted as G=(V, E), where V is the vertex set and E is the edge set.
V={1,2,3,4,5}, E={(1,1),(1,2),(1,4),(3,2)}

25. 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

26. Terms in graph strongly connected and loop-free
disconnected graph, components

27. complete graphs

28. 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

30. 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.

31. Examples of Poset
Ex 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 Define  on A={1, 2, 3, 4} by xy if x divide y. Then (A, ) is a poset.
Ex 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.

32. Original graph and Hasse diagram

33. 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 The relation on (a) is the subset relation, while the relations on the others are the divide relations.

34. totally ordered
Definition 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.

36. Topological sorting
Given a Hasse diagram for a partial order relation , how to find a total order  for which .

37. maximal and minimal
Definition 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, ).

38. Examples
Ex 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 How about the poset in Fig. 7.18? Do they have maximal or minimal elements?
Theorem If (A, ) is a poset and A is finite, then A has both a maximal and a minimal element.

39. Least and greatest
Definition 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.

40. Lower bound and upper bound
Definition 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).

41. Examples
Ex Let U={1, 2, 3, 4} with A=P(U) and let  be the subset relation on B. If B={{1}, {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 Let  be the “” relation on A. What are the results for the following cases?
A=R and B=[0, 1]
A=R and B={qQq2<2}
A=Q and B={qQq2<2}

42. Lattice
Definition The poset (A, ) is called a lattice if for all x, yA the elements lub{x, y} and glb{x, y} both exist in A.
Ex For A=N and x, yN, define xy by xy. Then lub{x, y}=max{x, y} and glb{x, y}=min{x, y}. (N,) is a lattice.
Ex For the poset (P(U), ), if S, TU, we have lub{S, T}=ST and glb{S, T}=ST and it is a lattice.

43. 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.

44. partition
Definition Given a set A and index set I, let AiA for iI. Then {Ai}iI is a partition of A if (a) A=iIAi and (b) AiAj= for ij.
Ex 7.52, A={1,…,10}….
Ex A partition of R

45. equivalence class
Definition the equivalence class of x, denoted [x], is defined by [x]={yAyx}
Ex Define the relation  on Z by xy if 4(x-y).
Ex Define the relation  on Z by ab if a2=b2.

46. 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 [x]=[y]; and (c) [x]=[y] or [x][y]=.
Ex 7.56.
Let A={1, 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 A=[1][2][4].
Consider an onto function f:A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). A=[1][4][2].
Ex If an equivalence relation  on A={1, 2, 3, 4, 5, 6, 7} induces the partition A={1, 2} {3}{4, 5, 7}{6}, what is ?

47. 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.

48. 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.

49. 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.

50. 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.

51. 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 s1Ek+1s2 if and only if v(s1, x)Ekv(s2,x) for all xI.

52. The procedure for the minimization
Set k=1. We determine the states that are 1-equivalent.
Having determined Pk, we determine the states that are (k+1)-equivalent. Note that if s1Eks2, then s1Ek+1s2 if and only if v(s1, x)Ekv(s2,x) for all xI.
If Pk+1=Pk, the process is completed.

53. Ex 7.60.
the original table in Table 7.1 and P1:{s1}, {s2, s5, s6}, {s3, s4}
Table 7.2. P2:{s1}, {s2, s5}, {s6}, {s3, s4}, P2=P3

54. refinement
Definition 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 P2<P1.
Theorem 7.9. In the minimization process, if Pk+1=Pk, then Pr+1=Pr for all rk+1.

55. distinguishing string
If s1Eks2 but s1Ek+1s2, then we have a string x=x1x2…xkxk+1Ik+1 such that w(s1, x)w(s2, x) but w(s1, x1x2…xk)=w(s2, x1x2…xk). We call this string as distinguishing string.
s1Ek+1s2x1I [v(s1, x1) Ek v(s2, x1)]

56. Ex 7.61
s2E1s6 but s2E2s6.
X=00 is the minimal distinguishing string for s2 and s6

57. Ex 7.62 s1 and s4 are 2 equivalent but are not 3-equivalent.
X=111 is the minimal distinguishing string for s1 and s4