Partial Order and Hasse Diagram

Which properties must a binary relation ⪯ on a set P satisfy for (P, ⪯) to be a poset (partially ordered set)?

a: Reflexive, Transitive, and Antisymmetric Explanation

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b: Reflexive and Symmetric Explanation

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c: Transitive and Antisymmetric Explanation

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d: Reflexive, Transitive, and Symmetric Explanation

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Which of the following best defines a maximal element

m \in P

in a poset

(P, ⪯)

a: There exists no element

a \in P

such that

a \prec m

(i.e., no element is strictly greater than

m

) Explanation

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b: There exists no element

a \in P

such that

m \prec a

(i.e., no element is strictly greater than

m

) Explanation

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c: There exists no element

a \in P

such that

a ⪯ m

(i.e., no element is at most

m

) Explanation

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m

is related to every other element in

P

Explanation

Let A={1,2,3,4,6,24,36,72}, Let ⪯ be the partial order defined by A ⪯ B if a divides b. Number of edges in the Hasse diagram of (A,⪯) is

a: 14 Explanation

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b: 10 Explanation

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c: 11 Explanation

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d: 13 Explanation

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Let

A = \{1, 2, 3, 6, 9, 18\}

and let

⪯

be the divisibility relation where

a ⪯ b

a

divides

b

. The number of edges in the Hasse diagram of

(A, ⪯)

a: 7 Explanation

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b: 5 Explanation

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c: 6 Explanation

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d: 8 Explanation

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Let P be the set of all people. Let R be a binary relation on P such that (a, b) is in R if a is a brother of b. Is R symmetric, transitive, an equivalence relation, a partial order relation?

a: NO, NO, NO, NO Explanation

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b: NO, NO, YES, NO Explanation

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c: NO, YES, NO, NO Explanation

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d: NO, YES, YES, NO Explanation

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Which of these is a valid Hasse diagram property?

a: Edges can cross each other for better visualization Explanation

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b: Elements are arranged vertically with larger elements above smaller ones Explanation

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c: All reflexive relations must be shown with self-loops Explanation

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d: Transitive edges must be explicitly drawn Explanation

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In the power set P({1,2,3}) ordered by set inclusion (⊆), how many incomparable pairs of elements are there?

a: 4 Explanation

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b: 6 Explanation

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c: 8 Explanation

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d: 12 Explanation

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Which statement about lattices is true?

a: Every finite partially ordered set is a lattice Explanation

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b: A lattice must have a greatest and least element Explanation

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c: Every two elements in a lattice must have both a join and a meet Explanation

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d: Elements in a lattice must form a total order Explanation

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Given a finite partially ordered set

(P, ⪯)

, which of the following statements is TRUE about the relationship between minimal elements and maximum-sized antichains?

a: The set of minimal elements is always a maximum-sized antichain Explanation

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b: Every maximum-sized antichain must contain all minimal elements Explanation

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c: The set of minimal elements forms an antichain, but it may not be of maximum size Explanation

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d: The number of minimal elements always equals the size of every maximum-sized antichain Explanation

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