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19.1: Motivation
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In many of the sets we encounter, there is some notion of elements being “less than or equal to” other elements in the set.
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19.2: Definition and properties
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Notice that in each of the examples in Section 19.1, the notion of “is smaller than” is defined via a relation.
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19.3: Graph for a partial order
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Hasse diagram: a diagram for the graph for a partial order on a finite set A, omitting reflexive loops and transitive “composite” edges, and placing “smaller” elements lower on the diagram instead of using arrows
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19.4: Total Orders
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Comparable Elements: elements a,b in a partially ordered set such that either a⪯b or b⪯a
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19.5: Maximal/minimal Elements
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Each of the following definitions are for a subset B of a partially ordered set A.
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19.6: Topological Sorting
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Sometimes we want to turn a partial order into a total order. What makes an order partial instead of total is the presence of pairs of incomparable elements.
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19.7: Activities
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19.8: Exercises
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