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Module #18:Relations- part II

Rosen 5th ed., ch. 7

§7.4: Closures of Relations

• For any property X, the “X closure” of a set A is defined as the “smallest” superset of A that has the given property.

• The reflexive closure of a relation R on A is obtained by adding (a,a) to R for each aA. I.e., it is R IA

• The symmetric closure of R is obtained by adding (b,a) to R for each (a,b) in R. I.e., it is R R−1

• The transitive closure or connectivity relation of R is obtained by repeatedly adding (a,c) to R for each (a,b),(b,c) in R.– I.e., it is

Zn

nRR*

Paths in Digraphs/Binary Relations

• A path of length n from node a to b in the directed graph G (or the binary relation R) is a sequence (a,x1), (x1,x2), …, (xn−1,b) of n ordered pairs in EG (or R).– An empty set of edges is considered a path of length 0

from a to a.– If any path from a to b exists, then we say that a is

connected to b. (“You can get there from here.”)• A path of length n≥1 from a to a is called a circuit

or a cycle.• Note that there exists a path of length n from a to

b in R if and only if (a,b)Rn.

Simple Transitive Closure Alg.

A procedure to compute R* of n elements with 0-1 matrices.

procedure transClosure(MR:rank-n 0-1 mat.)

A := B := MR;

for i := 2 to n beginA := A⊙MR; B := B A {join}

endreturn B {Alg. takes Θ(n4) time}

{note: A represents Ri}

Warshall’s Algorithm• Uses only Θ(n3) operations!

Procedure Warshall(MR : rank-n 0-1 matrix)

W := MR

for k := 1 to nfor i := 1 to n

for j := 1 to n

wij := wij (wik wkj)return W {this represents R*}

wij = 1 means there is a path from i to j going only through nodes ≤k

§7.5: Equivalence Relations

• An equivalence relation (e.r.) on a set A is simply any binary relation on A that is reflexive, symmetric, and transitive.– E.g., = itself is an equivalence relation.– For any function f:A→B, the relation “have the

same f value”, or = {(a1,a2) | f(a1)=f(a2)} is an equivalence relation, e.g., let m=“mother of” then = “have the same mother” is an e.r.

Equivalence Relation Examples

• “Strings a and b are the same length.”

• “Integers a and b have the same absolute value.”

• “Real numbers a and b have the same fractional part (i.e., a − b Z).”

• “Integers a and b have the same residue modulo m.” (for a given m>1)

Equivalence Classes

• Let R be any equiv. rel. on a set A.• The equivalence class of a,

[a]R :≡ { b | aRb } (optional subscript R)– It is the set of all elements of A that are “equivalent” to

a according to the eq. rel. R.– Each such b (including a itself) is called a

representative of [a]R.

• Since f(a)=[a]R is a function of a, any equivalence relation R be defined using aRb :≡ “a and b have the same f value”, given that f.

A set is divided into one or more classes

Equivalence Class Examples

• “Strings a and b are the same length.”– [a] = the set of all strings of the same length as a.

• “Integers a and b have the same absolute value.”– [a] = the set {a, −a}

• “Real numbers a and b have the same fractional part (i.e., a − b Z).”– [a] = the set {…, a−2, a−1, a, a+1, a+2, …}

• “Integers a and b have the same residue modulo m.” (for a given m>1)– [a] = the set {…, a−2m, a−m, a, a+m, a+2m, …}

Equivalent classes

• Let R be an equivalence relation on the set A, then the following statements are equivalent– aRb– [a] = [b]– [a] [b]

Partitions• A partition of a set A is the set of all the equivalence

classes {A1, A2, … } for some e.r. on A.

• The Ai’s are all disjoint and their union = A.

• They “partition” the set into pieces. Within each piece, all members of the set are equivalent to each other.

§7.6: Partial Orderings (POSET)

A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R)

Example

Let S = {1, 2, 3} and

let R = {(1,1), (2,2), (3,3), (1, 2), (3,1), (3,2) }

1

23

In a poset, the notation a b denotes that

This notation is used because the “less than or equal to” relation is a paradigm for a partial ordering. (Note that the symbol is used to denote the relation in any poset, not just the “less than or equals” relation.) The notation a b denotes thata b, but

( , )a b R

a b

Example

Let S = {1, 2, 3} and

let R = {(1,1), (2,2), (3,3), (1, 2), (3,1), (3,2)}

1

23

2 2 3 2

Comparable/Incomparable

The elements a and b of a poset (S, ) are called comparable if either a b or b a. When a and b are elements of S such that neither a b nor b a, a and b are called incomparable.

Example

Consider the power set of {a, b, c} and the subset relation. (P({a,b,c}), )

{ , } { , } and { , } { , }a c a b a b a c

So, {a,c} and {a,b} are incomparable

(S, )If is a poset and every two elements of S are comparable, S is called totally ordered or linearly ordered set, and is called a total order or a linear order. A totally ordered set is also called a chain.

If (A,R) is a poset, we say that A is totally ordered iffor all x,y in A either xRy or yRx. In this case R is called a total order.

Totally Ordered, Chains

(S, ) is a well-ordered set if it is a poset such that is a total ordering and such that every nonempty subset of S has a least element.

Well-Ordered Set

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1

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Hasse Diagrams

Given any partial order relation defined on a finite set, it is possible to draw the directed graph so that all of these properties are satisfied.

This makes it possible to associate a somewhat simpler graph, called a Hasse diagram, with a partial order relation defined on a finite set.

Hasse Diagrams (continued)

Start with a directed graph of the relation in which all arrows point upward. Then eliminate

1. the loops at all the vertices,

2. all arrows whose existence is implied by the transitive property,

3. the direction indicators on the arrows.

Example

Let A = {1, 2, 3, 9, 18} and consider the “divides” relation on A:

For all , , | for some integer .a b A a b b ka k

1

2

3

9

18

If aRb, normally a is located lower than b

Example

Eliminate the loops at all the vertices.

1

2

3

9

18

Eliminate all arrows whose existence is implied by the transitive property.Eliminate the direction indicators on the arrows.

1

2

3

9

18

a is a maximal in the poset (S, ) if there is nosuch that a b. Similarly, an element of a poset is called minimal if it is not greater than any element of the poset. That is, a is minimal if there is no element such that b a.

It is possible to have multiple minimals and maximals.

Maximal and Minimal Elements

b S

b S

a is the greatest element in the poset (S, ) if b a for all . Similarly, an element of a poset is called the least element if it is less than all other elements in the poset. That is, a is the least element if a b for all

Greatest Element, Least Element

b S

b S

a b

d

c

fe

y z

u w

x

v

Hasse diagrams

Upper bound, Lower bound

Sometimes it is possible to find an element that is greater than all the elements in a subset A of a poset (S, ). If u is an element of S such that a u for all elements , then u is called an upper bound of A. Likewise, there may be an element less than all the elements in A. If l is an element of S such that l a for all elements , then l is called a lower bound of A.

a A

a A

Least Upper Bound,Greatest Lower Bound

The element x is called the least upper bound of the subset A if x is an upper bound that is less than every other upper bound of A.

The element y is called the greatest lower bound of A if y is a lower bound of A and z y whenever z is a lower bound of A.

y z

u w

x

v

m n

p

Find the glb and lub of {y,m,u}, if they exist

The upper bounds are u,v. Since u v, u is the lub

The lower bounds are x,y. Since x y, y is the glb

How about glb and lub of {m,n,w}?

Lattices

A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice.

y z

u w

x

v

m n

p

y z

u w

x

v

m n

zu

w

x

v

m n

Are they lattices?

schedule

• 11/25 graph-part I

• 11/30 no class due to SNU entrance exam

• 12/2 graph-part II, tree

• 12/7 final exam