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Elementary Foundations: An introduction to topics in discrete mathematics

Jeremy Sylvestre

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Exercises 19.8 Exercises

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Recognizing a partial order from its graph.

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In each of Exercises 1–2, you are given a directed graph for a relation on the set A={a,b,c,d}. Determine whether the relation is a partial order. Justify your answers.
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Testing partial orders.

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In each of Exercises 3–6, you are given a set A and a relation R on A. Determine whether the relation is a partial order. Justify your answers.
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3.

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A is the set of all Augustana students; aRb means that student a has a higher GPA than student b.
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4.

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A is the power set of some finite set; SRT means |S|≀|T|.
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5.

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A is the set of words on some alphabet; wRwβ€² means |w|≀|wβ€²|, where |w| means the length of word w.
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6.

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A=RΓ—R; (x1,y1)R(x2,y2) means x1≀x2 and y1≀y2.
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Drawing Hasse diagrams.

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In each of Exercises 7–8, you are given a finite, partially ordered set A. Draw the Hasse diagram.
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8.

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A=Ξ£4βˆ—, the set of words of length 4 in the alphabet Ξ£={0,1}, under dictionary order.
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9.

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Draw all possible valid Hasse diagrams for each of the sets A={a,b} and B={a,b,c}. (Thus, you will have determined all possible partial orders on those sets.)
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10.

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Consider the β€œdivides” relation ∣ on N>0. Provide an example of a set AβŠ†N>0
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11.

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Let A={0,1,2}, and consider the partial order βŠ† on the power set P(A). List all pairs of incomparable elements in P(A).
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Determining maximal/maximum/minimal/minimum elements.

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In each of Exercises 12–16, you are given a partially ordered set A. Determine any and all maximal, maximum, minimal, and minimum elements.
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14.

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A=Nβˆ–{0,1} under the β€œdivides” relation ∣.
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15.

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A={2,5,11,13,22,65,110,143,496} under the β€œdivides” relation ∣.
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16.

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A is the set of the first ten prime numbers under the β€œdivides” relation ∣.
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17.

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Suppose βͺ― is a partial order on the set A={0,1,2} such that 1 is a maximal element. What are the possibilities for the Hasse diagram of βͺ―?
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Topological sorting.

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In each of Exercises 18–19, you are given the Hasse diagram for a partially ordered set A. Use the Topological sorting algorithm to determine a compatible total order on A.
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