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Lecture 4.5: POSets and Hasse Diagrams. CS 250, Discrete Structures, Fall 2011 Nitesh Saxena * Adopted from previous lectures by Cinda Heeren. Course Admin. HW4 has been posted Covers the chapter on Relations (lecture 4.*) Due at 11am on Nov 16 (Wednesday)

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Lecture 4.5: POSets and Hasse Diagrams

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## Lecture 4.5: POSets and Hasse Diagrams

CS 250, Discrete Structures, Fall 2011

NiteshSaxena

*Adopted from previous lectures by CindaHeeren

• HW4 has been posted

• Covers the chapter on Relations (lecture 4.*)

• Due at 11am on Nov 16 (Wednesday)

• Also has a 10-pointer bonus problem

Lecture 4.5 -- POSets and Hasse Diagrams

### Final Exam

• Thursday, December 8,  10:45am- 1:15pm, lecture room

• Our last lecture will be on December 6

• We plan to do a final exam review then

Lecture 4.5 -- POSets and Hasse Diagrams

### Outline

• Hasse Diagrams

• Some Definitions and Examples

• Maximal and miminal elements

• Greatest and least elements

• Upper bound and lower bound

• Least upper bound and greatest lower bound

Lecture 4.5 -- POSets and Hasse Diagrams

4

3

2

Draw edge (a,b) if a b

Don’t draw up arrows

Don’t draw self loops

Don’t draw transitive edges

1

### Hasse Diagrams

Hasse diagrams are a special kind of graphs used to describe posets.

Ex. In poset ({1,2,3,4}, ), we can draw the following picture to describe the relation.

Lecture 4.5 -- POSets and Hasse Diagrams

111

110

101

011

100

010

001

000

### Hasse Diagrams

Have you seen this one before? String comparison poset from last lecture

Lecture 4.5 -- POSets and Hasse Diagrams

### Maximal and Minimal

Reds are maximal.

Blues are minimal.

Consider this poset:

Lecture 4.5 -- POSets and Hasse Diagrams

### Maximal and Minimal: Example

Q: For the poset ({2, 4, 5, 10, 12, 20, 25}, |), what is/are the minimal and maximal?

A: minimal: 2 and 5

maximal: 12, 20, 25

Lecture 4.5 -- POSets and Hasse Diagrams

Did you get it right?

Intuition: If a is maxiMAL, then no one beats a. If a is maxiMUM, a beats everything.

Must minimum and maximum exist?

Only if set is finite.

No.

Only if set is transitive.

Yes.

### Least Element and Greatest Element

Definition: In a poset S, an element z is a minimum (or least) element if bS, zb.

Write the defn of maximum (geatest)!

Lecture 4.5 -- POSets and Hasse Diagrams

### Maximal and Minimal: Example

Q: For the poset ({2, 4, 5, 10, 12, 20, 25}, |), does the minimum and maximum exist?

A: minimum: [divisor of everything] No

maximum: [multiple of everything] No

Lecture 4.5 -- POSets and Hasse Diagrams

### A Property of minimum and maximum

Theorem: In every poset, if the maximum element exists, it is unique. Similarly for minimum.

Proof: Suppose there are two maximum elements, a1 and a2, with a1a2.

Then a1  a2, and a2a1, by defn of maximum.

Thus, our supposition was incorrect, and the maximum element, if it exists, is unique.

Similar proof for minimum.

Lecture 4.5 -- POSets and Hasse Diagrams

{a, b} has no UB.

### Upper and Lower Bounds

Defn: Let (S, ) be a partial order. If AS, then an upper bound for A is any element x  S (perhaps in A also) such that  a  A, a  x.

A lower bound for A is any x  S such that  a  A, x  a.

Ex. The upper bound of {g,j} is a. Why not b?

a

b

Ex. The upper bounds of {g,i} is/are…

A. I have no clue.

B. c and e

C. a

D. a, c, and e

c

d

e

f

j

h

g

i

Lecture 4.5 -- POSets and Hasse Diagrams

{g, h, i, j} has no LB.

### Upper and Lower Bounds

Defn: Let (S, ) be a partial order. If AS, then an upper bound for A is any element x  S (perhaps in A also) such that  a  A, a  x.

A lower bound for A is any x  S such that  a  A, x  a.

Ex. The lower bounds of {a,b} are d, f, i, and j.

a

b

Ex. The lower bounds of {c,d} is/are…

A. I have no clue.

B. f, i

C. j, i, g, h

D. e, f, j

c

d

e

f

j

h

g

i

Lecture 4.5 -- POSets and Hasse Diagrams

### Least Upper Bound and Greatest Lower Bound

Defn: Given poset (S, ) and AS, x  S is a leastupper bound (LUB) for A if x is an upper bound and for upper bound y of A, x  y.

x is a greatestlower bound (GLB) for A if x is a lower bound and if y  x for every lower bound y of A.

a

b

Ex. LUB of {i,j} = d.

Ex. GLB of {g,j} is…

A. I have no clue.

B. a

C. non-existent

D. e, f, j

c

d

e

f

j

h

g

i

Lecture 4.5 -- POSets and Hasse Diagrams

This is because c and d are incomparable.

### LUB and GLB

Ex. In the following poset, c and d are lower bounds for {a,b}, but there is no GLB.

Similarly, a and b are upper bounds for {c,d}, but there is no LUB.

a

b

c

d

Lecture 4.5 -- POSets and Hasse Diagrams

### Another Example

• What are the GLB and LUB, if they exist, of the subset {3, 9, 12} for the poset (Z+, |)?

• What are the GLB and LUB, if they exist, of the subset {1, 2, 4, 5, 10} for the poset (Z+, |)

Lecture 4.5 -- POSets and Hasse Diagrams

### Another Example

• What are the GLB and LUB, if they exist, of the subset {3, 9, 12} for the poset (Z+, |)?

LUB: [least common multiple] 36

GLB: [greatest common divisor] 3

• What are the GLB and LUB, if they exist, of the subset {1, 2, 4, 5, 10} for the poset (Z+, |)

LUB: [least common multiple] 20

GLB: [greatest common divisor] 1

Lecture 4.5 -- POSets and Hasse Diagrams

### Example to sum things up

For the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |), find the following:

• Maximal element(s)

• Minimal element(s)

• Greatest element, if it exists

• Least element, if it exists

• Upper bound(s) of {2, 9}

• Least upper bound of {2, 9}, if it exists

• Lowe bound(s) of {60, 72}

• Greatest lower bound of {60, 72}, if it exists

Lecture 4.5 -- POSets and Hasse Diagrams

### Example to sum things up

For the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |), find the following:

• Maximal element(s) [not divisors of anything] 27, 48, 60, 72

• Minimal element(s) [not multiples of anything] 2, 9

• Greatest element, if it exists [multiple of everything]No

• Least element, if it exists [divisor of everything]No

• Upper bound(s) of {2, 9} [common multiples]18, 36, 72

• Least upper bound of {2, 9}, if it exists [least common multiple]18

• Lower bound(s) of {60, 72} [common divisors]2, 4, 6, 12

• Greatest lower bound of {60, 72}, if it exists [greatest common divisor] 12

Lecture 4.5 -- POSets and Hasse Diagrams

### More Theorems

Theorem: For every poset, if the LUB for a set exist, it must be unique. Similarly for GLB.

Proof: Suppose there are two LUB elements, a1 and a2, with a1a2.

Then a1  a2, and a2a1, by defn of LUB.