Exercise 3.1 Poset Which of these are posets?
a) (Z,=) b) (Z,̸=) c) (Z,≥)
d) (Z,̸|) e) (R,=) f) (R,<) g) (R,≤) h) (R,̸=)
Partial Order
Answer these questions for the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |).
Exercise 3.2
a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element?
d) Is there a least element?
e) Find all upper bounds of {2, 9}.
f) Find the least upper bound of {2, 9}, if it exists.
g) Find all lower bounds of {60, 72}.
h) Find the greatest lower bound of {60, 72}, if it exists.
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Exercise 3.3 Lattice
Determine whether these posets are lattices.
a) ({1,3,6,9,12},1) b) ({1,5,25,125},|)
c) (Z,≥)
d) (P(S),⊇), where P(S) is the power set of a set S
Exercise 3.4 Chain and Antichain
a) Find a chain
b) Find an antichain
c) Find a maximal chain
d) Find a maximal antichain
Figure 1: problem 3.4
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Exercise 3.5 Cardinality
Give an example of two uncountable sets A and B such that A ∩ B is a) finite.
b) countably infinite.
c) uncountable.
Exercise 3.6 Cardinality
Show that there is no infinite set A such that |A| < |Z+| = א0.
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Reference
1. Rosen, Kenneth H., and Kamala Krithivasan. Discrete mathematics and its applica- tions: with combinatorics and graph theory. Tata McGraw-Hill Education, 2012.
2. https://services.math.duke.edu/lpereira/Teaching/ApCombSlides/3012_Lecture_18_handout.pd
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