Rosen, Discrete Mathematics and Its Applications, 6th edition
Extra Examples
Section 8.6—Partial Orderings
—Page references correspond to locations of Extra Examples icons in the textbook.
p.566, icon at Example 1
#1. Let A={(x, y)|x, y integers}. Define a relation Ron Aby the rule
(a, b)R(c, d)↔a≤cor b≤d.
Determine whether Ris a partial order relation on A.
Solution:
Ris reflexive: (a, b)R(a, b) for all elements (a, b) because a≤aor b≤bis always true.
Ris not antisymmetric: For example, (1,4)R(3,2) because 1 ≤3, and (3,2)R(1,4) because 2 ≤4. But
(1,4) =(3,2).
Ris not transitive: For example, (1,4)R(3,2) because 1 ≤3, and (3,2)R(0,3) because 2 ≤3. But
(1,4)R/ (0,3) because 1 ≤ 0and4≤ 3.
Therefore, Ris not a partial order relation because Ris neither antisymmetric nor transitive.
p.566, icon at Example 1
#2. Let A={(x, y)|x, y integers}. Define a relation Ron Aby the rule
(a, b)R(c, d)↔a=cor b=d.
Determine whether Ris a partial order relation on A.
Solution:
Ris reflexive: (a, b)R(a, b) for all elements (a, b) because a=aand b=bare always true.
Ris not antisymmetric: For example, (1,2)R(1,3) and (1,3)R(1,2) because 1 = 1, but (1,2) =(1,3).
Ris not transitive: For example, (1,2)R(1,3) because 1 = 1, and (1,3)R(4,3) because 3 = 3. But (1,2) =
(4,3) because 1 =4and2=3.
Therefore, Ris not a partial order relation because Ris neither antisymmetric nor transitive.
p.567, icon at Example 4
#1. Let Rbe the relation on the set of words in the English language where xRy if xprecedes (that is,
comes before) yin the dictionary. Show that Ris not a partial ordering.
Solution:
Note that Ris antisymmetric because if xprecedes yin the dictionary, where xand yare English words,
then ydoes not precede x.AlsonotethatRis transitive, for if xprecedes yin the dictionary and yprecedes
1
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