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Fall 2014 COMP 2300 Discrete Structures for Computation

Chapter 8.5 Partial Order Relations. Fall 2014 COMP 2300 Discrete Structures for Computation. Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University.

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Fall 2014 COMP 2300 Discrete Structures for Computation

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  1. Chapter 8.5 Partial Order Relations Fall 2014COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and PhysicsNorth Carolina Central University

  2. Let R be a relation on a set A. R is antisymmetric if, and only if, for all a and b in A, if and then a = b. • A relation R is NOT antisymmetricif and only if, there are elements a and b in A such that and but Antisymmetry Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  3. Example • Let and be the relations on {0, 1, 2} defined as follows: Draw the directed graphs for and and indicate which relations are antisymmetric. Testing for Antisymmetry of Finite Relations Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  4. Let be the “divides” relation on the set of all positive integers, and let be the “divides” relation on the set of all integers. • Is antisymmetric? Prove or give a counterexample. • Is antisymmetric? Prove or give a counterexample. Testing for Antisymmetry of “Divides” Relations Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  5. Let be the “divides” relation on the set of all positive integers, and let be the “divides” relation on the set of all integers. • Is antisymmetric? Prove or give a counterexample.Yes. Since • Is antisymmetric? Prove or give a counterexample.No. Since Testing for Antisymmetry of “Divides” Relations Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  6. Let R be a relation defined on a set A. R is a partial order relation if, and only if, R is reflexive, antisymmetric, and transitive. • R is an equivalence relation if, and only if, R is reflexive, symmetric, and transitive. Partial Order Relations Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  7. Let Abe any collection of sets and define the “subset relation”, , on A as follows: For all A ,Prove that is a partial order relation. A) is reflexive and transitive since It is also antisymmetric since The “Subset” Relation Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  8. Let | be the “divides” relation on a set A of positive integers. That is, for all ,Prove that | is a partial order relation on A. • Reflexive: and 1 is an integer. • Antisymmetric: True - Proved in Page 5. • Transitive: The “Divides” Relation on a Set of Positive Integers Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  9. Let S be a set of real numbers and define the “less than or equal to” relation, , on S as follows: For all real number x and y in S,Show that is a partial order relation. • Reflexive: • Antisymmetric: • Transitive: The “Less Than or Equal to” Relation Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  10. Let A be a set with a partial order relation R, and let S be a set of strings over A. Define a relation on S as follows: For any two strings in S, and , where m and n are positive integers, • If is the null string and s is any string in S, then If no strings are related other than by these three conditions, then is a partial order relation. : “x is less than or equal to y” Theorem 8.5.1 Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  11. The partial order relation of Theorem 8.5.1 is called the lexicographic order for S that corresponds to the partial order R on A. • Example • Let A={x, y} and let R be the following partial order relation on A:Let S be the set of all strings over A, and denoted by the lexicographic order for S that corresponds to R. • Is • Is • Is Lexicographic Order Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  12. The partial order relation of Theorem 8.5.1 is called the lexicographic order for S that corresponds to the partial order R on A. • Example • Let A={x, y} and let R be the following partial order relation on A:Let S be the set of all strings over A, and denoted by the lexicographic order for S that corresponds to R. • Is Yes. • Is Yes. • Is Yes. Lexicographic Order Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  13. Let A = {1, 2, 3, 9, 18} and consider the “divides” relation on A : For all a, b in A, the directed graph of this relation has the following appearance: Hasse Diagrams Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  14. For any directed graph for a partial order relation, we can construct a Hasse Diagram as follows: • Start with a directed graph of the relation, placing vertices on the page so that all arrows point upward. Then eliminate • The loops at all the vertices, • All arrows whose existence is implied by the transitive property, • The direction indicators on the arrows. Hasse Diagrams – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  15. Reconstructing a directed graph of a partial order relation R from a Hasse diagram Hasse Diagrams – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  16. Reconstructing a directed graph of a partial order relation R from a Hasse diagram Hasse Diagrams – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  17. Reconstructing a directed graph of a partial order relation R from a Hasse diagram Hasse Diagrams – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  18. Reconstructing a directed graph of a partial order relation R from a Hasse diagram Hasse Diagrams – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  19. Suppose is a partial order relation on a set A. Elements a and b of A are said to be comparable if, and only if, either a b or b a. Otherwise, a and b are called noncomparable. • Two Partial Order Relations • is comparable • is noncomparable. • If R is a partial order relation on a set A, and for any two elements a and b in A, either or , then R is a total order relation on A. Partially and Totally Ordered Sets Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  20. Let A be a set that is partially ordered with respect to a relation . A subset B of A is called a chain if, and only if, the elements in each pair of elements in B is comparable. In other words, a b or b a for all a and b in A. The length of a chain is one less than the number of elements in the chain. • Example • The set P({a, b, c}) is partially ordered with respect to the subset relation. Find a chain of length 3 in P({a, b, c}). • Since is a chain of length 3 in P({a, b, c}). A Chain of Subsets Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  21. Let a set A be partially ordered with respect to a relation . • An element a in A is called a maximal element of A if, and only if, for all b in A, either b a or b and a are not comparable. • An element a in A is called a greatest element of A if, and only if, for all b in A, b a. • An element a in A is called a minimal element of A if, and only if, for all b in A, either a bor b and a are not comparable. • An element a in A is called a least element of A if, and only if, for all b in A, a b. Maximal, greatest, minimal, and least elements Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  22. Let A = {a, b, c, d, e, f, g, h, i} have the partial ordering defined by the following Hasse diagram. Find all maximal, minimal, greatest, and least elements of A. Maximal element Greatest element No least element Maximal, greatest, minimal, and least elements – cont’ Minimal element Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  23. Given partial order relation and on a set A, is compatible with if, and only if, for all a and b in A, if a b then a b. • Given partial order relations and on a set A, is a topological sorting for if, and only if, is a total order that is compatible with . TopologicalSorting Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  24. Let be a partial order relation on a nonempty finite set A. To construct a topological sorting, • Pick any minimal element x in A. • Set • Repeat Steps a-c while • Pick any minimal element • Define x y. • Set and Constructing a Topological Sorting Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  25. Consider the set A = {2, 3, 4, 6, 18, 24} ordered by the “divides” relation |. The Hasse diagram of this relation is the following: Example Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  26. There are two minimal elements: 2 and 3. Pick one of them arbitrarily (i.e. 3). Then, the beginning of the total order is Total order: 3 Example – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  27. Now only minimal element remaining is 2. Total order: 3 2 Example – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  28. There are two minimal elements remaining: 4, 6. Pick 6. Total order: 3 2 6 Example – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

  29. We continue the process until nothing remains. Then we may haveTotal order: 3 2 6 18 4 24. Example – cont’ Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

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