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CSE 211- Discrete Structures1 Relations Ch 2 schaums, Ch 7 Rosen

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Hasse Diagram 2CSE 211- Discrete Structures

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Hasse Diagram Example 3CSE 211- Discrete Structures

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4 Hasse Diagrams Consider the graph for a finite poset ({1,2,3,4},) When we KNOW its a poset, we can simplify the graph 43214321 43214321 Called the Hassediagram CSE 211- Discrete Structures

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5 Hasse Diagram For the poset ({1,2,3,4,6,8,12}, |) CSE 211- Discrete Structures

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7 Greatest Element Least Element Maximal Element Minimal Element Upper Bound Lower Bound Greatest lower bound Least Upper bound

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CSE 211- Discrete Structures8 Find the lower and upper bounds of the subsets {a, b, c}, {j, h}, and {a, c, d, f}. The upper bounds of {a, b, c} are e, f, j, h and its lower bound is a. There is no upper bounds of {j, h}. And its lower bounds are a, b, c, d, e, f. The upper bounds of {a, c, d, f} are f, h and j. and its lower bound is a. Find the greatest lower bound and the least upper bound of {b, d, g}

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9 Lattices A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice.

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Section 7.6: Partial Orderings Def: A relation R on a set S is called a partial ordering (or partial order) if it is reflexive, antisymmetric, and transitive.

Section 7.6: Partial Orderings Def: A relation R on a set S is called a partial ordering (or partial order) if it is reflexive, antisymmetric, and transitive.

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