lecture 4.5: posets and hasse diagrams
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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
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
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
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
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