3.1 Relations

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU 1

Chapter 7 Relations: The Second Time Around

Relations: The Second Time

Around

Chapter 7

Equivalence Classes

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Chapter 7 Relations: The Second Time Around7.1 Relations Revisited: Properties of Relations

Ex. 7.3 Consider a finite state machine M =(S , I ,O,v,w).(a) For s1, s2 in S , define s1R s2 if v( s1, x)= s2 for some x in I .

Relation R establishes the first level of reachability .(b) The second level of reachability . s1R s2 if v( s1, x1 x2)= s2

for some x1 x2 in I 2. For the general reachability relation we

have v( s1, y)= s2 for some y in I *.(c) Given s1, s2 in S , the relation 1-equivalence , which is denoted

by s1E1 s2, is defined when w( s1, x)=w( s2, x) for all x in I . Thisidea can be extended to states being k-equivalence , where

s1Ek s2 if w( s1, y)=w( s2, y) for all y in I k .If two states are k -equivalent for all k in Z +, then they are calledequivalent .

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Def. 7.2 A relation R on a set A is called reflexive if for all x in A, ( x, x) is in R.

Ex. 7.4 For = {1,2,3,4}, a relation R will be reflexiveif and only if R {(1,1),(2,2),(3,3),(4,4)}. R = {( , )|

, , } is reflexive on .

Ex. 7.5 If | |= , | |= There are relations on .How many of these are reflexive?

We must include {( The remaining pairscan be either included or excluded from the relation. Therefore,

there are reflexive relations.

2

i

A A A x y

x y A x y A

A n A A n A

a a a A n n

n

i i

n n

.

, )| }.

2

2

2

2

2

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Def. 7.3 Relation R on set is call symmetric if ( , ) R ( , ) R, for all , . A x y y x x y A

Ex. 7.6 With A={1,2,3}, we have:(a) R 1={(1,2),(2,1),(1,3),(3,1)}, symmetric, but not reflexive;

(b) R 2={(1,1),(2,2),(3,3),(2,3)}, reflexive, but not symmetric;(c) R 3={(1,1),(2,2),(3,3)}, R 4={(1,1),(2,2),(3,3),(2,3),(3,2)},

both reflexive and symmetric;(d) R 5={(1,1),(2,3),(3,3)}, neither reflexive nor symmetric.

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To count the symmetric relations on = { writeas where and

contains12

subsets of

the form {( Therefore, there are

symmetric relations on . And reflexive andsymmetric relations on .

Def. 7.4 For a set , a relation R on is called if for all , , , ( ,

11 2

2

A a a a A A A A A a a i n A

a a i j n i j A n n

a a a a

A A

A A transitive x y z A x y

ni i

i j

i j j i nn n

n n

, , , },, {( , | }

{( , )| , , }. ( )

, ), ( , )}.( )

( )

21 2

2

1

212

1

1

2 2

2

2

2

), ( , ) R ( , ) R. y z x z

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Ex. 7.8 Define the relation R on the set Z + by aR b if a exactlydivides b. Then R is transitive, reflexive, but not symmetric (2R6,

but not 6R2)

Ex. 7.9 Consider the relation R on the set Z where R when0. Then R is reflexive, symmetric, but not transitive.

(3R0,0R - 7, but not 3R - 7)

Def. 7.5 Consider a relation R on a set , R is calledif for all , , ( R and R ) = .Ex. 7.11 the subset relation: R if , reflexive, transitive,antisymmetric, but not symmetric.Ex. 7.12 = {1,2,3}, R = {(1,2), (2,1),(2,3)} is neither symmetricnor antisymmetric. R = {(1,1), (2,2)} is both symmetric andantisymmetric.

a bab

A antisymmetrica b A a b b a a b

A B A B

A

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7/49Discrete Math by R.S. Chang, Dept. CSIE, NDHU 7


To count the antisymmetric relations on = { writeas where and

contains12

subsets of

the form {( There are 3 choices for this kind of

subsets: select one or none of them. Therefore, there are

antisymmetric relations on . Andreflexive and antisymmetric relations on .

11 2

2

A a a a A A A A A a a i n A

a a i j n i j A n n

a a a a

A A

ni i

i j

i j j i

n n n n n

, , , },, {( , | }

{( , )| , , }. ( )

, ), ( , )}.

( ) ( )

21 2

2

12

12

1

1

2 3 32 2

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Def. 7.6 A relation R os a set A is called a partial order , or a partial ordering relation , if R is reflexive, antisymmetric, andtransitive. (It is called a total order if for any a ,b in A, either a R b or bR a).

Examples of partial order: , , , ,

Ex. 7.15. Define the relation R on the set Z + by aR b if a exactlydivides b. R is a partial order.

Def. 7.7 An equivalence relation R on a set A is a relation that is

reflexive, symmetric, and transitive.Examples: aR b if a mod n=b mod n

The equality relation {(a i,a i)|a i in A} is both a partial order andan equivalence relation.

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Chapter 7 Relations: The Second Time Around7.2 Computer Recognition: Zero-One Matrices and Directed Graphs

Def. 7.8 If , , and are sets with R and R then the composite relation R R is a relation from todefined by R R and there are exists

with ( , ) R R

1 21 2

1 21 2

A B C A B B C A C x z x A z C

y B x y y z

,

{( , )| , ,, ( , ) }.

(Note the different ordering with function composition.)Ex. 7.17 A={1,2,3,4}, B={w, x, y, z }, and C={5,6,7}. Consider R 1={(1, x),(2, x),(3, y),(3, z )}, a relation from A to B, and R 2={(w,5),( x,6)}, a relation from B to C . Then R 1 R 2={(1,6),(2,6)}

is a relation from A to C .Theorem 7.1 Let A, B, C, and D be sets with R R and R Then R R R ( R R R

12 3 1 2 3

1 2 3

A B B C C D

,, . ( )

) .

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Def. 7.9 Given a set and a relation R on , we define theR recursively by (a) R R; and (b) for Z R

R R

Ex. 7.19 If = {1,2,3,4} and R = {(1,2), (1,3), (2,4), (3,2)}, thenR R and for 4, R

Def. 7.10 - (Use boolean operation, 1 +1 =1)

Ex. 7.20

1

is a 3 4 (0,1) - matrix.

1 + +1

2 3

A A power of n

An

zero one matrix

n

n

n

,.

{( , ), ( , ), ( , )}, {( , )}, .1 4 1 2 3 4 1 4

0 0 1

0 1 0 11 0 0 0

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Ex. 7.21 A={1,2,3,4}, B={w, x, y, z }, and C={5,6,7}.R 1={(1, x),(2, x),(3, y),(3, z )}, a relation from A to B, and R 2={(w,5),( x,6)}, a relation from B to C .

relation matrix M M

w

x y

z w x y z

M M M

( ) , ( )

( ) ( ) (

R R

5 6 7

R R R

1 2

1 2

1

23

4

0 1 0 0

0 1 0 00 0 1 1

0 0 0 0

1 0 0

0 1 00 0 0

0 0 0

0 1 0 00 1 0 0

0 0 1 1

0 0 0 0

1 0 00 1 0

0 0 0

0 0 0

0 1 00 1 0

0 0 0

0 0 0

1 2)R

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Let A be a set with | A|=n and R a relation on A. If M (R) is therelation matrix for R, then(a) M (R)= 0 (the matrix of all 0's) if and only if R is empty(b) M (R)= 1 (the matrix of all 1's) if and only if R= A A (c) M( Rm)=[ M (R)} m, for m in Z +.

Def. 7.11 Let = ( , = ( be two (0,1) -matrices. We say that precedes, or is less than, , and wewrite , if for all , .

Ex. 7.23 With = 10

00

11

and = 10

01

11

we have .

In fact, there are eight (0,1) - matrices for which .

E e F f m n E F

E F e f i m j n

E F E F

G E G

ij m n ij m n

ij ij

) )

,

,

1 1

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Def. 7.12 For Z is the (0,1) - matrix

whereif =

if Def 7.13 Let = ( be a (0,1) - matrix. The transpose of

, written , is the matrix ( wherefor all 1 , .

Ex. 7.24 =

+

ij

tr * *

tr

n I n ni j

i j A a

A A a a ai m j n

A A

n ij n n

ij m n

ji n m ji ij

, ( ),

,)

) ,

, .

1

0

10 1

0 01 1

0

1

0

0

1

1

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Theorem 7.2 Given a set with | |= , and a relation R on ,let denote the relation matrix for R. Then(a) R is reflexive if and only if

(b) R is symmetric if and only if =

(c) R is transitive if and only if =

(d) R is antisymmetric if and only if

tr

2

tr

A A n A M

I M

M M

M M M M

M M I

n

n

.

.

.

.( , )0 0 0 1 1 0 0 1 1 1

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Proof of (c) Let If ( , ), ( , ) R,

1 1 1 Hence ( , ) R.

Conversely, if R is transitive and is the relation matrix for R,

let be the entry in row and column of with

For to be 1 in there must exist at least one where

in . This happens only if ( , ), ( , ) R. WithR transitive, it then follows that ( , ) R.

2

2

2

M M x y y z

M x z

M

s x z M s

s M y A

m m M x y y z x z

xy yz xz

xz xz

xz

xy yz

.

.

, .

,

1

1 So and2

m

M M xz 1

.

7.2 Computer Recognition: Zero-One Matrices and Directed Graphs

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Def. 7.14 Directed Graphs G=(V , E ) (Digraph)

Ex. 7.25

1 2

4 3 5isolated vertex (node)

a loop V ={1,2,3,4,5} E ={(1,1),(1,2),(1,4),(3,2)}

1 is adjacent to 2.2 is adjacent from 1.

Undirected graphs: no direction in edges

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Ex. 7.26

( s1) b:=3;( s2) c:=b+2;( s3) a :=1;( s4) d :=a*b+5;( s5) e:=d -1;( s6) f :=7;( s7) e:=c+d ;( s8) g :=b* f ;

Construct a directed graph G=(V , E ), whereV ={ s1, s2, s3, s4, s5, s6, s7, s8} and ( s i, s j) in E if s i must be executed before s j.

s3

s1

s6

s4 s2 s8

s5 s7

precedence graph

precedence constraint scheduling3 processors: 3 time units2 processors: 4 time units

In general, n tasks,m processors: NP-completem=2: polynomialm=3: open problem

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Ex. 7.27 relations and digraphs

A={1,2,3,4}, R={(1,1),(1,2),(2,3),(3,2),(3,3),(3,4),(4,2)}

1

2

34

directed graphrepresentation

1

2

34

associatedundirected graph

a connected graph : a path exists between any two

vertices

cycle : a closed path (thestarting and ending vertices

are the same)

path : no repeated vertex

Def. 7.15 Strongly connected digrapha directed path exists between any two vertices

The above graph is not strongly connected. (no directed path from 3 to 1)

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Ex. 7.28 Components

1 2

3 4

two components

1 2

3 4

one component

Ex. 7.29 Complete Graphs: K nK

1K

2K

3 K 4K

5

(n vertices with n(n-1)/2 edges)

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directed graphs relations

adjacency matrices relation matrices

Ex. 7.30 If R is a relation on finite set A, then R is reflexive if

and only if its directed graph contains a loop at each vertex.Ex. 7.31 A relation R on a finite set A is symmetric if and onlyif its directed graph contains only loops and undirected edges.

Ex. 7.31 A relation R on a finite set A is transitive if and only if inits directed graph if there is a path from x to y, ( x, y) is an edge.A relation R on a finite set A is antisymmetric if and only if inits directed graph there are no undirected edges aside from loops.

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Ex. 7.33 A relation on a finite set A is an equivalence relation if and only if its associated (undirected) graph is one completegraph augmented by loops at every vertex or consists of thedisjoint union of complete graphs augmented by loops at everyvertex.

reflexive: loop on each vertexsymmetric: undirected edgetransitive: disjoint union of complete graphs

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Chapter 7 Relations: The Second Time Around7.3 Partial Orders: Hasse Diagrams

natural counting: N x+5=2 : Z 2 x+3=4 : Q

x2-2=0 : R x2+1=0 : C

Something was lost when we wentfrom R to C . We have lost the abilityto "order" the elements in C .

Let A be a set with R a relation on A. The pair ( A,R) is calleda partially ordered set , or poset , if relation R is a partial order.

Ex. 7.34 Let A be the courses offered at a college. Define R on A by xR y if x, y are the same course (reflexive) or if x is a prerequisitefor y. Then R makes A into a poset.

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Ex. 7.36 PERT (Performance Evaluation and Review Technique)

J 1 J 6

J 3

J 2 J 5

J 4

J 7

Find each job's earliest start time and latest start time.Those jobs which earliness equals to lateness are critical.All critical jobs form a critical path.

A poset

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not partial order

1 21

2

3

not antisymmetric not transitive or not antisymmetric

Ex. 7.37 Hasse diagram

1

2 3

4

1

2 3

4

a partialorder

corresponding Hasse diagram

Read bottom up .reflexivity andtransitive links

are not shown.

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Ex. 7.38{1,2,3}

{1,2} {1,3} {2,3}

{1} {2} {3}

subset relation exactly division relation

1

2

4

8

2 3 5 72 3 5 7 11

6

12385

35

equalityrelation

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Def. 7.16 If ( A,R) is a poset, we say that A is totally ordered if for all x, y in A either xR y or yR x. In this case R is called a total order .

For example, are total order for N,Z,Q,R. But partiallyordered in C.

But can we list the elements for a partially ordered set insome way?

sorting for a totally ordered set

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topological sorting for a partially ordered set

A

C

B E

G F D

Hasse diagram for a set of tasks

How to execute the tasks one at a timesuch that the partial order is not violated?

For example, BEACGFD, EBACFGD, ...

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topological sorting sequence (linear extension)

a b

c d

a^bc^d: 2 jumps

a^bd^c: 2 jumps

b^ac^d: 2 jumps

b^a^d^c: 3 jumps

bd^ ac: 1 jumps

Find a linear extensionwith minimum jumps.

(an NP-complete problem)

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Def. 7.17 If ( , R) is a poset, then an element of is calleda of if for all , ( , ) R.( R = ) An element is called a minimal element of

if whenever and , then ( , ) R ( R = ).

Ex 7.42 With R the "less than or equal to" relation on the set Z,(Z, ) is a poset with neither a maximal nor a minimal element.The poset (N, ), however, has a minimal element 0 but nomaximal element.

A x Aimal element A a A a x x a

x a x a y A A b A b y b y b y b y

max

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{1,2} {1,3} {2,3}

{1} {2} {3}

1 min

2

4

8 max

2 3 5 72 3 5 7 11

6

12385

35

Ex.7.43{1,2,3}max

min

max and min

min

max

unique max and min

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Theorem 7.3 If ( , R) is a poset and is finite, then has botha maximal and a minimal element.Proof: Let . If there is no element where and

R , then is maximal. Otherwise there is an elementwith and R If no element , satisfies

R , then is maximal. Otherwise we can find so thatwhile R and R Continuing in this

manner, since is finite, we get to an element with( R for any where

11 1

1 2 22 2

3 1 2 2 3

A A A

a A a A a aa a a a A

a a a a a A a aa a a a Aa a a a a a a a

A a Aa a a A a a

nn

12

2 13

3 2 1

. ,

, .

, ) n na, so is maximal.The proof for minimal element is similar.

In topological sorting, each time we find a maximal element,or each time we find a minimal element.

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Def 7.18 If ( , R) is a poset, then an element is calleda element if R for all . Element is called aelement if R for all .

Ex. 7.44 Let = {1,2,3} and R be the subset relation.

(a) With = ( ), the poset ( , ) has as a least elementand as a greatest element.(b) For = the collection of nonempty subsets of , the poset ( , ) has as a greatest element. There is no leastelement.

(c) For = the collection of proper subsets of , the poset ( , ) has as a least element. There is no greatestelement

A x Aleast x a a A y A greatest a y a A

U

A P U AU B U

B U

C U C

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Theorem 7.4 If the poset ( A,R) has a greatest (least) element,then that element is unique.Proof: Suppose that x, y in A and that both are greatest elements.Then ( x, y) and ( y, x) are both in R. As R is antisymmetric, itfollows that x= y. The proof for the least element is similar.

Def. 7.19 Let ( , R) be a poset with . An element iscalled a ( ) for if R ( R ) for all .An element ' is called a , , (

, ) if it is a lower bound (upper bound) of andfor all other lower bounds (upper bounds) " of we have

"R ' ( 'R ").

A B A x Alower upper bound B x b b x b B

x A greatest lower bound least upper bound B

x B x x x x

glblub

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Ex. 7.46 Let U ={1,2,3,4}, with A=P( U ), let R be the subsetrelation on A. If B={{1},{2},{1,2}}, then {1,2}, {1,2,3},{1,2,4}, and {1,2,3,4} are all upper bounds for B(in A),whereas {1,2} is a least upper bound (in B). Meanwhile, agreatest lower bound for B is the empty set , which is not in B.

Ex. 7.47 Let R be the "less than or equal to" relation for the poset( A, R). (a) If A=R and B=[0,1] or [1,0) or (0,1] or (0,1), then Bhas glb 0 and lub 1. (b) If A=R , B={q in Q |q2

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Theorem 7.5 If ( ,R) is a poset and , then has at mostone lub (glb).

Def 7.20 The poset ( ,R) is called a if for any ,the element lub{ , } and glb{ , } both exist in .

Ex. 7.48 For = N and , N, define R by . Then{ , } = { , }, { , } = { , }, and (N, )

is a lattice.

Ex. 7.49 ( ( ), ) is a lattice with lub{ , } = and{ , } = for , .

A B A B

A lattice x y A x y x y A

A x y x y x y x y x y x y x y

P U S T S T S T S T S T U

lub max glb min

glb

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Chapter 7 Relations: The Second Time Around7.4 Equivalence Relations and Partitions

Def. 7.21 Given a set and an index set , let for each . Then { is a partition of if (a) = and (b) for all , where .

Each subset is called a or of the partition.

Ex. 7.51 = {1,2,3,4,5,6,7,8,9,10}, then each of the followingdetermines a partition of :(a) b)c)

11

A I A Ai I A A A A A A i j I i j

A cell block

A A

A A A A A A i i

ii i I

ii I

i j

i

i

},

{ , , , , }, { , , , , }( { , , }, { , , , }, { , , }( { ,

1 2 3 4 5 6 7 8 9 101 2 3 4 6 7 9 5 8 10

5

22 3

}, .

[ , ).}

1 5

1

i

A i A i i A

ii i

Ex. 7.52 = R, and for each Z, let Then{ is a partition of R.Z

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Def. 7.22 Let R be an equivalence relation on a set . For any , the equivalence class of , denoted is defined by R

Ex. 7.53 R if 4| ( - ) for , Z. For this equivalence wefind that 0 Z

1 Z2 Z3

A x A x x x y A y x

x y x y x yk k

k k k k

,{ | }.

{ , , , , , , } { | }{ , , , , , , } { | }{ , , , , , , } { | }{ , , , , , , } {

8 4 0 4 8 47 31 5 9 4 16 2 2 6 10 4 25 1 3 7 11

4 3

1 2 3k k Z | }.

, , , }And { 0 provides a partition of Z.

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Ex. 7.54 For , Z, R if Then R is an equivelencerelation. What can we say about the corresponding partition of

Z? In general, forn any Z Therefore,

Z = [ ]

Theorem 7.6 If R is an equivalence relation on a set , and, , then (a) [ ]; (b) R if and only if and

(c) [ ] = [ ] or [ ] [ ] = .

2

+

=

a b a b a b

n n n n n

n

A x y A x x x y x y

x y x y

n

2

0

.

, { , }.

.

;

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Ex. 7.58 If an equivalence relation R on A={1,2,3,4,5,6,7} inducesthe partition ? What is R? A { , } { } { , , } { }1 2 3 4 5 7 6R = ({1,2} {1,2}) ({3} {3}) {(4,5,7} {4,5,7})

({6} {6}), |R|= 2 2

1 3 1 152 2 2 .

Theorem 7.7 If A is a set, then (a) any equivalence relation R on A induces a partition of A, and (b) any partition of A gives rise to anequivalence relation on A.

Theorem 7.8 For any set A, there is a one-to-one correspondence between the set of equivalence relations on A and the set of partitions of A.

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Ex. 7.59 (a) If = {1,2,3,4,5,6}, how many relations on areequivalence relations?one - to - one correspondence between equivalence relationsand partitions

( , )(b) How many of the equivalence relations satisfy 1,2 4

1,2,4 are in the same partition. ( , )

=

A A

S i

S i

i

i

6 203

4 15

1

6

1

4.

.

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Chapter 7 Relations: The Second Time Around7.5 Finite State Machines: The Minimization Process

Given s1, s2 in S , the relation 1-equivalence , which is denoted by s1E1 s2, is defined when w( s1, x)=w( s2, x) for all x in I . Thisidea can be extended to states being k-equivalence , where

s1Ek s2 if w( s1, y)=w( s2, y) for all y in I k .

If two states are k -equivalent for all k in Z +, then they are calledequivalent , denoted by s1E s2. Hence our objective is to determinethe partition of S induced by the equivalence relation E and select one state for each equivalence class .

h l h d d

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observations:(1) If two states in a machine are not 2-equivalent, could they possibly be 3-equivalent?

No. If and are not 2 - equivalent, then there exists in

such that ( , ) ( So for any ,(

= (

1 22 s s xy

I w s xy w s xy z I w s xyz w s xy w v s xy z w s xy w v s xy z

w s xyz

1 21 1 1 2 2

2

, )., ) ( , ) ( ( , ), ) ( , ) ( ( , ), )

, ).

In general, to find states that are ( k +1)-equivalent, we look atstates that are k -equivalent.

Ch 7 R l i Th S d Ti A d

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observations:(b) Suppose E We wish to determine whether EThat is does ( ( for all strings

Because E E w(sConsequently, ( ( if

( ( ( ( That is, if ( E

In general, for s

2 1 3

2 1 1

2

s s s sw s x x x w s x x x

x x x I s s s s x w s xw s x x x w s x x x

w v s x x x w v s x x xv s x v s x

1 2 21 1 2 3 2 1 2 3

1 2 33

1 2 1 2 1 2 1

1 1 2 3 2 1 2 31 1 2 3 2 1 2 31 1 2 1

. ?, ) , )

? , , ) ( , )., ) , )

, ), ) , ), )., ) ( , ).

1 2 1 +1

, s we have E if and only if (i) E and (ii) ( E for all .

S s s s s v s x v s x x I

k k k

1 22 1 2, ) ( , )


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s1 s4 s3 0 1 s2 s5 s2 1 0 s3 s2 s4 0 0 s4 s5 s3 0 0 s5 s2 s5 1 0 s6 s1 s6 1 0

Ex. 7.60

0 1 0 1v w step 1: determine 1-equivalent states byexamining outputs

P 1: { s1},{ s2, s5, s6},{ s3, s4}

input 0

s5 s2 s1 s2 s5

P 2: { s1},{ s2, s5},{ s6},{ s3, s4}

input 1 s2 s5 s4 s3

P 3= P 2, the process is complete with 4 states.

A

C

B

D


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Could it be that P 3= P 2, but P 3= P 4?

Def. 7.23 If are any partitions of a set , thenis called a refinement of and we write if everycell of is contained in a cell of When and

we write This occurs when at least one cellin is properly contained in a cell in

In the minimization process, because ( + 1) -equivalence implies - equivalence. So each successive

partition refines the preceding partition.

2 21

2 1 22 22 1

+

P P A P P P P

P P P P P P P P

P P

P P k k

k k

12 1

11 1

1

,, ,

..

.


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Theorem 7.9 In applying the minimization process, if 1and and are partitions with thenfor any + 1.Proof: If not, let ( + 1) be the smallest subscript such that

Then so there exist with E but E But E E for all .And with we then find that E for all . So

+ + +

+ + 1

+ 1

k P P P P P P

r k r k

P P P P s s S s s s s s s v s x v s x x I

P P v s x v s x x I

k k k k r r

r r r r r

r r r r r r

1 1 1

1 1 1 2 2

1 1 2 2 1 1 21 1 2

,

. , ,. ( , ) ( , )

, ( , ) ( , ) s s P P r r r 1 + +E Consequently,1 2 1. .

If there must be a smallest integer 0 such that

E but E That is, there is an such that( (but they will agree on the first outputs)is called a for and

1

1 1+

1 2

s s k

s s s s x I w s x w s x k x distinguishing string s s

k k k

22 1 2

1

1 2

,

., ) ( , ).

.

Ch t 7 R l ti Th S d Ti A d

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How to find the minimal length distinguishing string?

s1

s4

s3 0 1 s2 s5 s2 1 0

s3 s2 s4 0 0 s4 s5 s3 0 0s

5 s 2 s 5 1 0 s6 s1 s6 1 0

0 1 0 1v w

Ex. 7.61 P 1: { s1},{ s2, s5, s6},{ s3, s4}

input 0

s5 s2 s1 s2 s5

P 2: { s1},{ s2, s5},{ s6},{ s3, s4}

input 1

s2 s5 s4 s3

A distinguishing string for s2 and s6 is 00 (or 01).


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s1 s4 s2 0 1

s2 s5 s2 0 0 s3 s4 s2 0 1 s4 s3 s5 0 1 s5 s2 s3 0 0

0 1 0 1v w P 1: { s1 , s3, s4},{ s2, s5}

input 1 s2 s2 s5 s2 s3

input 1 s2 s2 s5

Ex. 7.62

P 2: { s1 , s3, s4},{ s2},{ s5}

P 3: { s1 , s3},{ s4},{ s2},{ s5}

distinguishing string for s1 and s4: 111


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Exercise: P318:10P330:20P340:26P346:10P356:14

Date post:	03-Apr-2018
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