Transcript Chapter 7 Relations : the second time around

Chapter 7
Relations : the
second time
around
Yen-Liang Chen
Dept of Information Management
National Central University
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 .
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.
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 kequivalent for all k.
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
n
2
2
have AA=n , so there are
relations
on A. Among them, 2( n n ) is reflexive.
2
2
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={(1,
3={(1,
4={(1,
5={(1,
2),
1),
1),
1),
1),
(2,
(2,
(2,
(2,
(2,
1),
2),
2),
2),
3),
(1,
(3,
(3,
(2,
(3,
3), (3, 1)}
3), (2, 3)}
3)}
3), (2, 3), (3, 2)}
3)}
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}.
n
(1 / 2 )( n 2 n )
• So, we have totally 2  2
symmetric
relations on A.
• If the relations are both symmetric and reflexive,
2
we have 2(1/ 2)( n n )
choices.
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?
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 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)}
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.
2
– So, we have totally 2 n  3(1/ 2)( n n ) 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?
partial order relation
• Definition 7.6. A relation  is called a
partial order, if  is reflexive, antisymmetric and transitive.
• Ex 7.15. Define the relation  on the
set Z+ by ab if a divides b.
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)}
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.
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?
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
the power of relation
• Definition 7.9. We define the powers
of relation  by (a) 1=; (b)
n+1= n
• Ex 7.19. If ={(1, 2), (1, 3), (2, 4),
(3, 2)}, then what is 2 and 3 and
4.
matrix representation
• A relation can be represented by an mn zeroone matrix.
• Ex 7.17. Consider 1={(1, x), (2, x), (3, y), (3,
z)} and 2={(w, 5), (x, 6)}. What is 12?
matrix representation
• Ex 7.19. If ={(1, 2), (1, 3), (2, 4), (3,
2)}, then what is 2 and 3 and 4.
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
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.
1 0 1
E

0 0 1
1 0 1
F

0 1 1
Identity matrix
I2=
1 0 
0 1 


I 3=
1 0 0 
0 1 0 


0 0 1 
Transpose of a matrix
0 1 
A=  0 0 


1 1 
Atr=
0 0 1
1 0 1


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.
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.
• V={1,2,3,4,5}, E={(1,1),(1,2),(1,4),(3,2)}
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
Terms in graph
• strongly connected and loop-free
• disconnected graph, components
complete graphs
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
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.
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 A={1, 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.
Original graph and Hasse
diagram
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.
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.
Topological sorting
• Given a Hasse diagram for a partial
order relation , how to find a total
order  for which .
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 R a. Similarly, y is a minimal
element of A if for all bA, byb R 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, ).
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.
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.
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).
Examples
• Ex 7.47. 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 7.48. 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}
Lattice
• Definition 7.20. 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 7.49. 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 7.50. 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.
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.
partition
• Definition 7.21. 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 7.53. A partition of R
equivalence class
• Definition 7.22. the equivalence class
of x, denoted [x], is defined by
[x]={y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 a2=b2.
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 7.58. 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 ?
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.
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.
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.
k-equivalence
• For the states S, we define the kequivalence 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.
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.
The procedure for the
minimization
1. Set k=1. We determine the states
that are 1-equivalent.
2. 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.
3. If Pk+1=Pk, the process is completed.
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
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 P2<P1.
• Theorem 7.9. In the minimization
process, if Pk+1=Pk, then Pr+1=Pr for
all rk+1.
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)]
Ex 7.61
• s2E1s6 but s2E2s6.
• X=00 is the minimal distinguishing string
for s2 and s6
Ex 7.62
• s1 and s4 are 2 equivalent but are not 3equivalent.
• X=111 is the minimal distinguishing
string for s1 and s4