28-PartialOrders

28-PartialOrders - Discussion #28 Partial Orders Discussion...

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Discussion #28 Chapter 5, Section 4.8 1/13 Discussion #28 Partial Orders
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Discussion #28 Chapter 5, Section 4.8 2/13 Topics Weak and strict partially ordered sets (posets) Total orderings Hasse diagrams Bounded and well-founded posets
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Discussion #28 Chapter 5, Section 4.8 3/13 Partial Orders Total orderings: single sequence of elements Partial orderings: some elements may come before/after others, but some need not be ordered Examples of partial orderings: foundation framing plumbing wiring finishing {a, b, c} {a, b} {a, c} {b, c} {a} {b} {c} “must be completed before” “set inclusion,
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Discussion #28 Chapter 5, Section 4.8 4/13 Partial Order Definitions (Poset Definitions) A relation R: S S is called a ( weak ) partial order if it is reflexive, antisymmetric, and transitive. 1 2 3 A relation R: S S is called a strict partial order if it is irreflexive , antisymmetric, and transitive. 1 2 3 e.g. on the integers e.g. < on the integers
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Discussion #28 Chapter 5, Section 4.8 5/13 Total Order A total ordering is a partial ordering in which every element is related to every other element. (This forces a linear order or chain.) Examples: R: on {1, 2, 3, 4, 5} is total. Pick any two; they’re related one way
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28-PartialOrders - Discussion #28 Partial Orders Discussion...

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