9-Partial Orderings

9-Partial Orderings - EECS 210 Discrete Structures David O...

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Partial Orderings David O. Johnson EECS 210 (Fall 2016) 1 EECS 210 Discrete Structures David O. Johnson Fall 2016

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Reminders Connect 2 due: 11:59 PM, Thursday, September 29 Homework 3 due: Thursday, October 6 before your lecture Partial Orderings David O. Johnson EECS 210 (Fall 2016) 2
Any Questions? Partial Orderings 3 David O. Johnson EECS 210 (Fall 2016)

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Partial Orderings (Section 9.6) Partial Orderings and Partially-ordered Sets (posets) Comparability Lexicographic Orderings Hasse Diagrams Maximal and Minimal Elements Greatest & Least Elements Upper & Lower Bounds Lattices Topological Sorting Partial Orderings 4 David O. Johnson EECS 210 (Fall 2016)
What are Partial Orderings? We often use relations to order some or all of the elements of sets. For instance, we order words using the relation containing pairs of words ( x,y ), where x comes before y in the dictionary. We schedule projects using the relation consisting of pairs ( x, y ), where x and y are tasks in a project such that x must be completed before y begins. We order the set of integers using the relation containing the pairs ( x, y ), where x is less than y. When we add all of the pairs of the form ( x, x ) to these relations, we obtain a relation that is reflexive , antisymmetric , and transitive . These are properties that characterize relations used to order the elements of sets. Partial Orderings David O. Johnson EECS 210 (Fall 2016) 5

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Partial Orderings Definition : A relation R on a set S is called a partial ordering, or partial order, if it is reflexive, antisymmetric, and transitive. A set together with a partial ordering R is called a p artially o rdered set , or poset , and is denoted by ( S , R ). Members of S are called elements of the poset. Partial Orderings 6 David O. Johnson EECS 210 (Fall 2016)
Partial Orderings ( continued ) Example : Show that the “greater than or equal” relation ( ) is a partial ordering on the set of integers. Reflexivity : a a for every integer a . Antisymmetry : If a b and b a , then a = b. Transitivity : If a b and b c , then a c. These properties all follow from the order axioms for the integers. ( See Appendix 1 ). Partial Orderings 7 David O. Johnson EECS 210 (Fall 2016)

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Partial Orderings ( continued ) Example : Show that the inclusion relation ( ) is a partial ordering on the power set of a set S . Reflexivity : A A whenever A is a subset of S . Antisymmetry : If A and B are positive integers with A B and B A , then A = B . Transitivity : If A B and B C , then A C . The properties all follow from the definition of set inclusion. Partial Orderings 8 David O. Johnson EECS 210 (Fall 2016)
Comparability Definition : The elements a and b of a poset ( S , ) are comparable if either a b or b a .

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