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Elementary Foundations: An introduction to topics in discrete mathematics

Jeremy Sylvestre

Activities 19.7 Activities
A4US

Activity 19.1.

Let F⊆N represent the set of all divisors of 30. Let A={a,b,c}.
Note: In Task c you will compare your work from Task a and Task a, so keep your work!

(a)

Draw the Hasse diagram for the subset partial order ⊆ on P(A).

(b)

Draw the Hasse diagram for the “divides” partial order ∣ on F.

(c)

Compare your two Hasse diagrams. Can you devise a function f:F→P(A) that would deserve to be called an order-preserving correspondence between F and P(A)?

Activity 19.3.

Let A={a,b,c,d,e}. Carry out the following steps for each of the scenarios below.
  1. Draw the Hasse diagram for a partial order on A with the requested features.
  2. In your diagram, identify all maximal/minimal elements.
  3. Identify all pairs of incomparable elements.

(a)

A has both a maximum and a minimum.

(b)

A has a maximum but no minimum.

(c)

A has a minimum but no maximum.

(d)

A has neither a maximum nor a minimum.

Activity 19.4.

Suppose ⪯ 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 ⪯?

Activity 19.5.

Using the proper strategy for proving uniqueness (see Procedure 6.10.2), prove that if a partially ordered set A has a maximum element, then that element is the unique maximum element.
How can your proof be modified to show that a minimum element is also unique?

Activity 19.6.

Recall that (a,b)⊆R means an open interval on the real number line:
(a,b)={x∈R|a<x<b}.
Let ≤ be the usual “less than or equal to” total order on the set
A=(−2,0)∪(0,2).
Consider the subset
B={−1n|n∈N,n≥1}⊆A.
Determine an upper bound for B in A. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound.
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