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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) Also has a 10-pointer bonus problem Please start early Lecture 4.5 -- POSets and Hasse Diagrams Final Exam Thursday, December 8, 10:45am1:15pm, 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 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. 4 3 2 1 b 1. Draw edge (a,b) if a 2. Don’t draw up arrows 3. Don’t draw self loops 4. Don’t draw transitive edges Lecture 4.5 -- POSets and Hasse Diagrams Hasse Diagrams Have 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 Minimal Consider this poset: Reds are maximal. Blues are minimal. 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 Least Element and Greatest Element Definition: In a poset S, an element z is a minimum (or least) element if bS, zb. Intuition: If a is maxiMAL, then no one beats a. If a is maxiMUM, a beats everything. Write the defn of maximum (geatest)! Did you get it right? Must minimum and maximum exist? A. Only if set is finite. B. No. C. Only if set is transitive. D. Yes. 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} a c g b d e f h i j is a. Why not b? Ex. The upper bounds of {g,i} is/are… A. I have no clue. {a, b} has no B. c and e UB. C. a D. a, c, and e 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} a c g b d e f h i j are d, f, i, and j. Ex. The lower bounds of {c,d} is/are… A. I have no clue. B. f, i {g, h, i, j} C. j, i, g, h has no LB. D. e, f, j 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. 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 c g b d e f h i j 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 Lecture 4.5 -- POSets and Hasse Diagrams 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 This is because c and d are incomparable. 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: 1. 2. 3. 4. 5. 6. 7. 8. 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: 1. 2. 3. 4. 5. 6. 7. 8. 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 Today’s Reading Rosen 9.6 Lecture 4.5 -- POSets and Hasse Diagrams