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Rosen, Section 8.5 Equivalence Relations. Longin Jan Latecki Temple University, Philadelphia [email protected] Some slides from Aaron Bloomfield. Introduction. Certain combinations of relation properties are very useful

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Rosen section 8 5 equivalence relations l.jpg

Rosen, Section 8.5Equivalence Relations

Longin Jan Latecki

Temple University, Philadelphia

[email protected]

Some slides from Aaron Bloomfield


Introduction l.jpg
Introduction

  • Certain combinations of relation properties are very useful

  • In this set we will study equivalence relations:A relation that is reflexive, symmetric and transitive

  • Next slide set we will study partial ordering:A relation that is reflexive, antisymmetric, and transitive

  • The difference is whether the relation is symmetric or antisymmetric


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Outline

  • What is an equivalence relation

  • Equivalence relation examples

  • Related items

    • Equivalence class

    • Partitions


Slide4 l.jpg

We can group properties of relations together to define new types of important relations.

_________________

Definition: A relation R on a set A is an equivalence relationiff R is

• reflexive

• symmetric

• transitive

Two elements related by an equivalence relation are called equivalent.

  • Consider relation R = { (a,b) | len(a) = len(b) }, where len(a) means the length of string a

    • It is reflexive: len(a) = len(a)

    • It is symmetric: if len(a) = len(b), then len(b) = len(a)

    • It is transitive: if len(a) = len(b) and len(b) = len(c), then len(a) = len(c)

    • Thus, R is a equivalence relation


Equivalence relation example l.jpg
Equivalence relation example types of important relations.

  • Consider the relation R = { (a,b) | a ≡ b (mod m) }

    • Remember that this means that m | a-b

    • Called “congruence modulo m”

  • Is it reflexive: (a,a)  R means that m | a-a

    • a-a = 0, which is divisible by m

  • Is it symmetric: if (a,b)  Rthen (b,a)  R

    • (a,b) means that m | a-b

    • Or that km = a-b. Negating that, we get b-a = -km

    • Thus, m | b-a, so (b,a)  R

  • Is it transitive: if (a,b)  R and (b,c)  R then (a,c)  R

    • (a,b) means that m | a-b, or that km = a-b

    • (b,c) means that m | b-c, or that lm = b-c

    • (a,c) means that m | a-c, or that nm = a-c

    • Adding these two, we get km+lm = (a-b) + (b-c)

    • Or (k+l)m = a-c

    • Thus, m divides a-c, where n = k+l

  • Thus, congruence modulo m is an equivalence relation


Rosen section 8 5 question 1 l.jpg
Rosen, section 8.5, question 1 types of important relations.

  • Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack

  • { (0,0), (1,1), (2,2), (3,3) }

    • Has all the properties, thus, is an equivalence relation

  • { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) }

    • Not reflexive: (1,1) is missing

    • Not transitive: (0,2) and (2,3) are in the relation, but not (0,3)

  • { (0,0), (1,1), (1,2), (2,1), (2,2), (3,3) }

    • Has all the properties, thus, is an equivalence relation

  • { (0,0), (1,1), (1,3), (2,2), (2,3), (3,1), (3,2) (3,3) }

    • Not transitive: (1,3) and (3,2) are in the relation, but not (1,2)

  • { (0,0), (0,1) (0,2), (1,0), (1,1), (1,2), (2,0), (2,2), (3,3) }

    • Not symmetric: (1,2) is present, but not (2,1)

    • Not transitive: (2,0) and (0,1) are in the relation, but not (2,1)


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Rosen, Section 8.5, question 9 types of important relations.

  • Suppose that A is a non-empty set, and f is a function that has A as its domain. Let R be the relation on A consisting of all ordered pairs (x,y) where f(x) = f(y)

    • Meaning that x and y are related if and only if f(x) = f(y)

  • Show that R is an equivalence relation on A

  • Reflexivity: f(x) = f(x)

    • True, as given the same input, a function always produces the same output

  • Symmetry: if f(x) = f(y) then f(y) = f(x)

    • True, by the definition of equality

  • Transitivity: if f(x) = f(y) and f(y) = f(z) then f(x) = f(z)

    • True, by the definition of equality


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Rosen, Section 8.5, question 11 types of important relations.

  • Show that the relation R, consisting of all pairs (x,y) where x and y are bit strings of length three or more that agree except perhaps in their first three bits, is an equivalence relation on the set of all bit strings

  • Let f(x) = the bit string formed by the last n-3 bits of the bit string x (where n is the length of the string)

  • Thus, we want to show: let R be the relation on A consisting of all ordered pairs (x,y) where f(x) = f(y)

  • This has been shown in question 9 on the previous slide


Slide9 l.jpg

An types of important relations.equivalence class of an element x:

[x] = {y | <x, y> is in R}

[x] is the subset of all elements related to [x] by R.

The element in the bracket is called a representative

of the equivalence class. We could have chosen any one.

Theorem: Let R be an equivalence relation on A. Then either

[a] = [b] or [a] ∩[b] = Φ

The number of equivalence classes is called the rank of theequivalence relation.

Let A={a,b,c} and R be given by a digraph:


More on equivalence classes l.jpg
More on equivalence classes types of important relations.

  • Consider the relation R = { (a,b) | a mod 2 = b mod 2 } on the set of integers

    • Thus, all the even numbers are related to each other

    • As are the odd numbers

  • The even numbers form an equivalence class

    • As do the odd numbers

  • The equivalence class for the even numbers is denoted by [2] (or [4], or [784], etc.)

    • [2] = { …, -4, -2, 0, 2, 4, … }

    • 2 is a representative of its equivalence class

  • There are only 2 equivalence classes formed by this equivalence relation


More on equivalence classes11 l.jpg
More on equivalence classes types of important relations.

  • Consider the relation R = { (a,b) | a = b or a = -b }

    • Thus, every number is related to additive inverse

  • The equivalence class for an integer a:

    • [7] = { 7, -7 }

    • [0] = { 0 }

    • [a] = { a, -a }

  • There are an infinite number of equivalence classes formed by this equivalence relation


Slide12 l.jpg

Theorem: types of important relations.Let R be an equivalence relation on a set A.

The equivalence classes of R partition the set A into disjoint nonempty subsets whose union is the entire set.

This partition is denoted A/Rand called

• the quotient set, or

• the partition of A induced by R, or,

• A modulo R.

Definition: Let S1, S2, . . ., Snbe a collection of subsets of a set A. Then the collection forms a partition of A if the subsets are nonempty, disjoint and exhaust A:

Note that { {}, {1,3}, {2} } is not a partition (it contains the empty set).

{ {1,2}, {2, 3} } is not a partition because ….

{ {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3.


Slide13 l.jpg

It is easy to recognize equivalence relations using digraphs:

• The equivalence class of a particular element forms a universal relation (contains all possible edges) between the elements in the equivalence class.

The (sub)digraph representing the subset is called a complete (sub)digraph, since all arcs are present.

Example: All possible equivalence relations on a set A with 3 elements:


Rosen section 8 5 question 44 l.jpg
Rosen, section 8.5, question 44 digraphs:

  • Which of the following are partitions of the set of integers?

  • The set of even integers and the set of odd integers

    • Yes, it’s a valid partition

  • The set of positive integers and the set of negative integers

    • No: 0 is in neither set

  • The set of integers divisible by 3, the set of integers leaving a remainder of 1 when divided by 3, and the set of integers leaving a remaineder of 2 when divided by 3

    • Yes, it’s a valid partition

  • The set of integers less than -100, the set of integers with absolute value not exceeding 100, and the set of integers greater than 100

    • Yes, it’s a valid partition

  • The set of integers not divisible by 3, the set of even integers, and the set of integers that leave a remainder of 3 when divided by 6

    • The first two sets are not disjoint (2 is in both), so it’s not a valid partition


Slide15 l.jpg

1. Determine whether the relations represented by these zero-one matricesare equivalence relations. If yes, with how many equivalence classes?

2. What are the equivalence classes (sets in the partition) of the integersarising from congruence modulo 4?

3. Can you count the number of equivalence relations on a set A with n elements. Can you find a recurrence relation?

The answers are

• 1 for n = 1

• 2 for n = 2

• 5 for n = 3

How many for n = 4?


Slide16 l.jpg

Theorem ( zero-one matricesBell number)

Let p(n) denotes the number of different equivalence relations on a set with n elements

(which is equivalent to the number of partitions of the set with n elements). Then

p(n) is called Bell number, named in honor of Eric Temple Bell

Examples:

p(0)=1, since there is only one partition of the empty set:

into the empty collection of subsets

p(1)=C(0,0)p(0)=1, since {{1}} is the only partition of {1}

p(2)=C(1,0)p(1)+C(1,1)p(0)=1+1=5, since portions of {1,2} are {{1,2}} and {{1},{2}}

p(3)=5, since, the set { 1, 2, 3 } has these five partitions.

{ {1}, {2}, {3} }, sometimes denoted by 1/2/3.

{ {1, 2}, {3} }, sometimes denoted by 12/3.

{ {1, 3}, {2} }, sometimes denoted by 13/2.

{ {1}, {2, 3} }, sometimes denoted by 1/23.

{ {1, 2, 3} }, sometimes denoted by 123.


Slide17 l.jpg

Proof zero-one matrices(Bell number):

We want to portion {1, 2, …, n}.

For a fixed j, A is a subset of j elements from {1, 2, …, n-1} union {n}.

Note that j can have values from 0 to n-1.

We can select a subset of j elements from {1, 2, …, n-1} in C(n-1,j) ways,

and we have p(n-1-j) partitions of the remaining n-1-j elements. ■


Slide18 l.jpg

Theorem: zero-one matricesIf R1and R2are equivalence relations on A, then R1∩R2is an equivalence relation on A.

Proof: It suffices to show that the intersection of

• reflexive relations is reflexive,

• symmetric relations is symmetric, and

• transitive relations is transitive.


Slide19 l.jpg

Definition: zero-one matricesLet R be a relation on A.

Then the reflexive, symmetric, transitive closure of R, tsr(R), is an

equivalence relation on A, called the equivalence relation induced by R.

Example:


Slide20 l.jpg

Theorem: zero-one matricestsr(R) is an equivalence relation.

Proof:

We need to show that tsr(R) is still symmetric and reflexive.

• Since we only add arcs vs. deleting arcs when computing closures it must be that tsr(R) is reflexive since all loops <x, x> on the diagraph must be present when constructing r(R).

• If there is an arc <x, y> then the symmetric closure of r(R) ensures there is an arc <y, x>.

• Now argue that if we construct the transitive closure of sr(R) and we add an edge <x, z> because there is a path from x to z, then there must also exist a path from z to x (why?) and hence we also must add an edge <z, x>.

Hence the transitive closure of sr(R) is symmetric.



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Quick survey zero-one matrices

  • I felt I understood the material in this slide set…

  • Very well

  • With some review, I’ll be good

  • Not really

  • Not at all


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Quick survey zero-one matrices

  • The pace of the lecture for this slide set was…

  • Fast

  • About right

  • A little slow

  • Too slow


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Quick survey zero-one matrices

  • How interesting was the material in this slide set? Be honest!

  • Wow! That was SOOOOOO cool!

  • Somewhat interesting

  • Rather boring

  • Zzzzzzzzzzz


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