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
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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)
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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)
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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)
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Comparability Definition : The elements a and b of a poset ( S , ) are comparable if either a b or b a .
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