Discrrete mathematics for computer science digraphs and relations

Digraphs and Relations

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Walks in digraph G

walk from u to v and
from v to w
u

v
w

implies walk from u to w
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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
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Walks in digraph G

transitive relation R:
u R v AND v R w
IMPLIES u R w
G is transitive
+

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transitivity
Theorem:

R is a transitive if
R = G+ for some

digraph G
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Transitive Closure

G+ is the

transitive closure
of the binary
relation G
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reflexivity
relation R on set A
is reflexive if
a R a for all a A
≤ on numbers and ⊆
on sets are reflexive
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reflexivity
For any digraph G,
*
G is reflexive

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Reflexive Transitive
Closure

G* is the reflexive

transitive closure
of the binary
relation G
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two-way walks
If there is a walk from
u to v and a walk back
from v to u then u and v
are strongly connected.

uG v
*

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AND

vG u
*

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symmetry
relation R on set A
is symmetric if
a R b IMPLIES b R a
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Paths in DAG D

path from u to v implies
no path from v to u
unless u=v
*
u D v and u≠v
IMPLIES NOT(v D
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*

u)
12

antisymmetry
antisymmetric relation R:
u R v IMPLIES NOT(v R u)
for any u ≠ v
If D is a DAG then
*

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(weak) partial orders

Reflexive, Transitive,
and Antisymmetric
examples:

•⊆ is (weak) p.o. on sets
•  is (weak) p.o. on 
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weak partial
orders

Theorem:

R is a WPO if
R = D* for some

DAG D
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equivalence
relations

transitive,
symmetric &
reflexive
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equivalence
relations

Theorem:

R is an equiv rel if
R = the strongly
connected relation
of some digraph

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equivalence relation
examples:

• = (equality)
• same size
• sibling (same
parents)
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Equivalence Relation
An equivalence relation decomposes the
domain into subsets called equivalence
classes where aRb if a and b are in the
same equivalence class
In the digraph of an equivalence relation,
all the members of an equivalence
class are reachable from each other but
not from any other equivalence class

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Graphical Properties of
Relations
Reflexive

Symmetric

Transitive Equivalence Relation

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Finis

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