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Lecture 4.5: POSets and Hasse Diagrams

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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

Course Admin

- 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
- Please start early

Lecture 4.5 -- POSets and Hasse Diagrams

Final Exam

- Thursday, December 8, 10:45am- 1:15pm, lecture room
- Heads up!
- Please mark the date/time/place

- 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

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 DiagramsHasse 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

110

101

011

100

010

001

000

Hasse DiagramsHave 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

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 ElementDefinition: In a poset S, an element z is a minimum (or least) element if bS, zb.

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 a1a2.

Then a1 a2, and a2a1, by defn of maximum.

So a1=a2, a contradiction.

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

Upper and Lower Bounds

Defn: Let (S, ) be a partial order. If AS, 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

Upper and Lower Bounds

Defn: Let (S, ) be a partial order. If AS, 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 AS, 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 GLBEx. 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 a1a2.

Then a1 a2, and a2a1, by defn of LUB.

So a1=a2, a contradiction.

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

Similar proof for GLB.

Lecture 4.5 -- POSets and Hasse Diagrams