Lecture 4.5: POSets and Hasse Diagrams

CS 250, Discrete Structures, Fall 2011

Nitesh Saxena

*Adopted from previous lectures by Cinda Heeren

Lecture 4.5 -- POSets and Hasse Diagrams

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

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Our last lecture will be on December 6 We plan to do a final exam review then

Lecture 4.5 -- POSets and Hasse Diagrams

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

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.

1. Draw edge (a,b) if a b

2. Don’t draw up arrows

3. Don’t draw self loops

4. Don’t draw transitive edges

4

3

2

1

Lecture 4.5 -- POSets and Hasse Diagrams

Lecture 4.5 -- POSets and Hasse Diagrams

Hasse DiagramsHave you seen this one before? String

comparison poset from last lecture

111

110 101 011

100 010 001

000

Lecture 4.5 -- POSets and Hasse Diagrams

Maximal and MinimalConsider this poset: Reds are maximal.

Blues are minimal.

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

Lecture 4.5 -- POSets and Hasse Diagrams

Least Element and Greatest Element

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

Write the defn of maximum (geatest)!

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?

A. Only if set is finite.B. No.C. Only if set is transitive.D. Yes.

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

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 BoundsDefn: 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.

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

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

a b

d

jf

ih

e

c

g

Ex. The upper bounds of {g,i} is/are…A. I have no clue.B. c and eC. aD. a, c, and e

{a, b} has no UB.

Lecture 4.5 -- POSets and Hasse Diagrams

Upper and Lower BoundsDefn: 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.

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

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

a b

d

jf

ih

e

c

g

Ex. The lower bounds of {c,d} is/are…A. I have no clue.B. f, iC. j, i, g, hD. e, f, j

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

Lecture 4.5 -- POSets and Hasse Diagrams

Least Upper Bound and Greatest Lower Bound

Defn: Given poset (S, ) and AS, x S is a least upper bound (LUB) for A if x is an upper bound and for upper bound y of A, x y.

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

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

d

jf

ih

e

c

g

Ex. GLB of {g,j} is…A. I have no clue.B. aC. non-existent D. e, f, j

Lecture 4.5 -- POSets and Hasse Diagrams

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

dc

This is because c and d are

incomparable.

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] 36GLB: [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] 20GLB: [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:1.Maximal element(s)2.Minimal element(s)3.Greatest element, if it exists4.Least element, if it exists5.Upper bound(s) of {2, 9}6.Least upper bound of {2, 9}, if it exists7.Lowe bound(s) of {60, 72}8.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:1.Maximal element(s) [not divisors of anything] 27, 48, 60, 722.Minimal element(s) [not multiples of anything] 2, 93.Greatest element, if it exists [multiple of everything] No4.Least element, if it exists [divisor of everything] No5.Upper bound(s) of {2, 9} [common multiples] 18, 36, 726.Least upper bound of {2, 9}, if it exists [least common multiple] 187.Lower bound(s) of {60, 72} [common divisors] 2, 4, 6, 128.Greatest lower bound of {60, 72}, if it exists [greatest common divisor] 12

Lecture 4.5 -- POSets and Hasse Diagrams

Lecture 4.5 -- POSets and Hasse Diagrams

More TheoremsTheorem: 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

Today’s Reading Rosen 9.6