Maggie Johnson
Handout #9
CS103B
Relations
Key topics:
* Introduction and Definitions
* Graphs and Relations
* Properties of Relations
* Equivalence Relations
* Partial Orderings
* Composition of Relations
* Matrix Representation
* Closures
* Topological Sorting
* PERT and CPM; Scheduling
Suppose we have a set of Greek deities:
G
= { Zeus, Apollo, Cronus, Poseidon }
As everyone knows, Zeus is the father of Apollo, Cronus is the father of Poseidon, and Cronus is
also the father of Zeus.
It seems that there exists some combination of the elements of
G
that satisfy
the "is the father of" relation.
To express this more precisely:
A
relation
R
between two sets
A
and
B
is a subset of
A
X
B
.
In other words,
A
X
B
produces a set
of ordered pairs <
a
,
b
>,
a
coming from set
A
, and
b
from set
B
.
Some of these pairs will be
"interesting", i.e., will satisfy our relation.
For these pairs, we can write
a
R
b
, where
R
is the
symbol for our relation.
Alternatively, we can write <
a
,
b
>
∈
R
. (Note that
R
is a generic relation
symbol.
Some relations, such as "lessthan" have their own symbol:
<.)
To be precise, what we have defined is a
binary relation
, so called because it operates on ordered
pairs.
We can also define
unary relations,
which operate on single elements, or ternary relations,
which operate on ordered triples.
In general an
nary relation
will operate on ntuples.
Example 1
Let's consider the “is the father of” relation (which we will denote by
F
) on the set
G
X
G
.
We can
figure out that
G
X
G
= { <Zeus, Zeus>, <Zeus, Apollo>, <Zeus, Cronus>,
<Zeus, Poseidon>, <Apollo, Zeus>, <Apollo, Apollo>,
<Apollo, Cronus>, <Apollo, Poseidon>, <Cronus, Zeus>,
<Cronus, Apollo>, <Cronus, Cronus>, <Cronus, Poseidon>,
<Poseidon, Zeus>, <Poseidon, Apollo>,
<Poseidon, Cronus>, <Poseidon, Poseidon> }
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View Full DocumentBut of the set
G
X
G
, only a subset satisfies the "is the father of" relation.
Thus, applying the
F
relation to
G
X
G
. yields the set:
{ <Zeus, Apollo>, <Cronus, Poseidon>, <Cronus, Zeus> }
meaning that Zeus
F
Apollo,
Cronus
F
Poseidon, and Cronus
F
Zeus.
Graphs and Relations
Graphs are a general representation for expressing manytomany relationships.
The easiest way to
understand the definition of a graph is to look at a picture.
The following are all examples of graphs:
Intuitively, we get the idea that a graph is a bunch of points connected by lines.
The formal
definition conveys this concept a bit more obtusely:
A
graph
is an ordered triple <
N
,
A
,
f >
where
N
is a nonempty set of
nodes
or
vertices
(dots)
A
is a set of
arcs
or
edges
(lines)
f
is a function associating each arc
a
with an
un
ordered pair
x
,
y
of nodes
called the
endpoints
of
a
.
Graphs are incredibly useful structures, and the first use we will put them to is to represent the
family relation described by the “father of” relation.
C
r o
n
u
s
Z e u
s
P
o
s e i d
o
n
A
p
o
l l o
The astute reader may note that this graph has arrows rather than lines connecting the nodes.
That is
because the graph above is actually a special type of graph called a directed graph.
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 Winter '08
 SAHAMI,M
 Order theory, Binary relation, Transitive relation, relation, Partially ordered set

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