# Ch9.6-Partial-Orders.ppt - Partial Orders Jorge A Cobb The...

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1 Partial Orders Jorge A. Cobb The University of Texas at Dallas

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2 Partially ordered sets Partial orders (relations that are reflexive, antisymmetric, and transitive) can be used to order the elements of a set Equivalence relations, on the other hand, break a set into pieces (subsets) of “same” or equal elements (equiv. classes) A set S together with a partial order R is called a partially ordered set or poset ( S , R ).
3 Example posets The set of reals with “≤” The set of positive integers with “|” (divides) The power set of a set with “ The set of English words with “lexicographic order” on strings compare successive letters until a different one is found – order of these letters determines the order of strings

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4 More examples Is “<“ and the reals a PoSet? Consider set S = {all people}, R = {(x,y) | (x is older than y}. Is R a partial order (and (S,R) a PoSet?) R is antisymmetric (if I am older than you you can’t be older than me) R is transitive R is not reflexive though.
5 Comparable elements Let be a partial order on a set S. Elements x S and y S are called comparable if x y or y x . Otherwise, x and y are called incomparable . Note also that if x is comparable with y , then y is comparable with x

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6 a is comparable with n, m, r and s x is comparable with m, r, and s s is comparable with r, m, a, and x a and x are NOT comparable m and n are NOT comparable n and x are not comparable a r x s m n Let R be the reflexive and transitive closure of this graph (R is thus a partial order)
7 Total order (linear order) A partial order on a set S is called a total order or a linear order if any two elements are comparable. In that case, the set S is called a totally ordered or linearly ordered set, or a chain .

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8 All elements are comparable with each other (transitive edges are removed for ease of drawing)
9 Examples Is ≤ on the set of reals a total order? Yes. For any reals x and y , either x y or y x . | on the set of positive integers? No. Neither 2 divides 3, nor 3 divides 2.

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• Fall '19
• Order theory, Partially ordered set, Transitivity, antisymmetric, Jorge A. Cobb

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