Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 8.5 Partial Order Relations

Antisymmetry 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 antisymmetric if and only if, there are elements a and b in A such that and but 2 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Testing for Antisymmetry of Finite Relations 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. a. b. 3 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Testing for Antisymmetry of “Divides” Relations Let be the “divides” relation on the set of all positive integers, and let be the “divides” relation on the set of all integers. a.Is antisymmetric? Prove or give a counterexample. b.Is antisymmetric? Prove or give a counterexample. 4 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Testing for Antisymmetry of “Divides” Relations Let be the “divides” relation on the set of all positive integers, and let be the “divides” relation on the set of all integers. a.Is antisymmetric? Prove or give a counterexample. Yes. Since b.Is antisymmetric? Prove or give a counterexample. No. Since 5 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Partial Order Relations 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. 6 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

The “Subset” Relation Let A be 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 7 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

The “Divides” Relation on a Set of Positive Integers 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. 1.Reflexive: and 1 is an integer. 2.Antisymmetric: True - Proved in Page 5. 3.Transitive: 8 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

The “Less Than or Equal to” Relation 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. 1.Reflexive: 2.Antisymmetric: 3.Transitive: 9 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Theorem 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. 10 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University : “ x is less than or equal to y ”

Lexicographic Order The partial order relation of Theorem 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. a.Is b.Is c.Is 11 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Lexicographic Order The partial order relation of Theorem 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. a.Is Yes. b.Is Yes. c.Is Yes. 12 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Hasse Diagrams 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: 13 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Hasse Diagrams – cont’ 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 1.The loops at all the vertices, 2.All arrows whose existence is implied by the transitive property, 3.The direction indicators on the arrows. 14 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Hasse Diagrams – cont’ Reconstructing a directed graph of a partial order relation R from a Hasse diagram 15 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Hasse Diagrams – cont’ Reconstructing a directed graph of a partial order relation R from a Hasse diagram 16 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Hasse Diagrams – cont’ Reconstructing a directed graph of a partial order relation R from a Hasse diagram 17 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Hasse Diagrams – cont’ Reconstructing a directed graph of a partial order relation R from a Hasse diagram 18 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Partially and Totally Ordered Sets 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. 19 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

A Chain of Subsets 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 }). 20 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Maximal, greatest, minimal, and least elements Let a set A be partially ordered with respect to a relation. 1.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. 2.An element a in A is called a greatest element of A if, and only if, for all b in A, b a. 3.An element a in A is called a minimal element of A if, and only if, for all b in A, either a b or b and a are not comparable. 4.An element a in A is called a least element of A if, and only if, for all b in A, a b. 21 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Maximal, greatest, minimal, and least elements – cont’ 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. 22 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Maximal element Greatest element Minimal element No least element

Topological Sorting 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. 23 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Constructing a Topological Sorting Let be a partial order relation on a nonempty finite set A. To construct a topological sorting, 1.Pick any minimal element x in A. 2.Set 3.Repeat Steps a-c while a.Pick any minimal element b.Define x y. c.Set and 24 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Example Consider the set A = {2, 3, 4, 6, 18, 24} ordered by the “divides” relation |. The Hasse diagram of this relation is the following: 25 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Example – cont’ 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 26 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Example – cont’ Now only minimal element remaining is 2. Total order: Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Example – cont’ There are two minimal elements remaining: 4, 6. Pick 6. Total order: Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Example – cont’ We continue the process until nothing remains. Then we may have Total order: Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University