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,=)̸
Exercise 3.2 Partial Order
Answer these questions for the poset ({2,4,6,9,12, 18,27,36,48,60,72},|).
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.
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
Figure 1: problem 3.4
a) Find a chain
b) Find an antichain
c) Find a maximal chain
d) Find a maximal antichain
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.
