Mathematics for Computer Science
MIT 6.042J/18.062J
Partial Orders &
Equivalence
Relations
Albert R Meyer
2011
October 21,
lec7F.1
Walks in digraph G
walk from u to v and
from v to w
v
u
w
implies walk from u to w
Albert R Meyer
2011
October 21,
lec7F.2
Walks in digraph G
walk from u to v and
from v to w, implies
walk from u to w:
+
+
u G v AND v G w
+
IMPLIES u G w
Albert R Meyer
2011
October 21,
lec7F.3
Walks in digraph G
transitive relation R:
u R v AND v R w
IMPLIES u R w
+
G is transitive
Albert R Meyer
2011
October 21,
lec7F.4
transitivity
Theorem:
R is a transitive iff
R = G+ for some
digraph G
Albert R Meyer
2011
October 21,
lec7F.5
Paths in DAG D
pos length path from
u to v implies
no path from v to u
+
+
u D v IMPLIES NOT(v D u)
Albert R Meyer
2011
October 21,
lec7F.6
Paths in DAG D
asymmetric relation R:
u R v IMPLIES NOT(v R u)
+
D
is asymmetric
Albert R Meyer
2011
October 21,
lec7F.7
strict partial orders
transitive &
asymmetric
Albert R Meyer
2011
October 21,
lec7F.8
strict partial orders
examples:
•
on sets
• “indirect prerequisite” on
MIT subjects
• less than, <, on real
numbers
Albert R Meyer
2011
October 21,
lec7F.9
strict partial orders
Theorem:
R is a SPO iff
R = D+ for some
DAG D
Albert R Meyer
2011
October 21,
lec7F.10
weak partial orders
same as a strict partial
order R, except that
a R a always holds
examples:
•⊆ is weak p.o. on sets
•  is weak p.o. on
Albert R Meyer
2011
October 21,
lec7F.12
reflexivity
relation R on set A
is reflexive iff
a R a for all a A
*
G is reflexive
Albert R Meyer
2011
October 21,
lec7F.13
antisymmetry
binary relation R is
antisymmetric iff
it is asymmetric
except for a R a case.
Albert R Meyer
2011
October 21,
lec7F.14
antisymmetry
antisymmetric relation R:
u R v IMPLIES NOT(v R u)
for u v
*
D
is antisymmetric
Albert R Meyer
2011
October 21,
lec7F.15
weak partial orders
transitive,
antisymmetric &
reflexive
Albert R Meyer
2011
October 21,
lec7F.16
weak partial orders
Theorem:
R is a WPO iff
R = D* for some
DAG D
Albert R Meyer
2011
October 21,
lec7F.17
two-way walks
walk from u to v and
back from v to u:
u and v are strongly
connected.
*
u G v AND
Albert R Meyer
2011
October 21,
*
vG u
lec7F.18
symmetry
relation R on set A
is symmetric iff
a R b IMPLIES b R a
Albert R Meyer
2011
October 21,
lec7F.19
equivalence relations
transitive,
symmetric &
reflexive
Albert R Meyer
2011
October 21,
lec7F.20
equivalence relations
Theorem:
R is an equiv rel iff
R = the strongly
connected relation
of some digraph
Albert R Meyer
2011
October 21,
lec7F.21
equivalence relation
examples:
•
•
•
•
=(equality)
≡ (mod n)
same size
same color
Albert R Meyer
2011
October 21,
lec7F.22
Graphical Properties of Relations
Reflexive
Asymmetric
NO
Transitive
Albert R Meyer
2011
Symmetric
October 21,
lec7F.23
Graphical Properties of Relations
path-total
(looks like a path/chain)
Albert R Meyer
2011
October 21,
lec7F.24
Team Problems
Problems
1−5
Albert R Meyer
2011
October 21,
lec7F.35