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Relations and Partial Orders
A relation is a mathematical tool for describing associations between elements of
sets. Relations are widely used in computer science, especially in databases and
scheduling applications. A relation can be defined across many items in many sets,
but in this text, we will focus on
binary
relations, which represent an association
between two items in one or two sets.
7.1
Binary Relations
7.1.1
Definitions and Examples
Definition 7.1.1.
Given sets
A
and
B
, a
binary relation
R
W
A
!
B
from
1
A
to
B
is a subset of
A
±
B
. The sets
A
and
B
are called the
domain
and
codomain
of
R
, respectively. We commonly use the notation
aRb
or
a
²
R
b
to denote that
.a; b/
2
R
.
A relation is similar to a function. In fact, every function
f
W
A
!
B
is a rela
tion. In general, the difference between a function and a relation is that a relation
might associate multiple elements of
B
with a single element of
A
, whereas a func
tion can only associate at most one element of
B
(namely,
f .a/
) with each element
a
2
A
.
We have already encountered examples of relations in earlier chapters. For ex
ample, in Section
5.2
, we talked about a relation between the set of men and the
set of women where
mRw
if man
m
likes woman
w
. In Section
5.3
, we talked
about a relation on the set of MIT courses where
c
1
Rc
2
if the exams for
c
1
and
c
2
cannot be given at the same time. In Section
6.3
, we talked about a relation on the
set of switches in a network where
s
1
Rs
2
if
s
1
and
s
2
are directly connected by a
wire that can send a packet from
s
1
to
s
2
. We did not use the formal definition of a
relation in any of these cases, but they are all examples of relations.
As another example, we can define an “inchargeof” relation
T
from the set of
MIT faculty
F
to the set of subjects in the 2010 MIT course catalog. This relation
contains pairs of the form
.
h
instructorname
i
;
h
subjectnum
i
/
1
We also say that the relationship is
between
A
and
B
, or
on
A
if
B
D
A
.
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Chapter 7
Relations and Partial Orders
(Meyer,
6.042),
(Meyer,
18.062),
(Meyer,
6.844),
(Leighton,
6.042),
(Leighton,
18.062),
(Freeman,
6.011),
(Freeman,
6.881)
(Freeman,
6.882)
(Freeman,
6.UAT)
(Eng,
6.UAT)
(Guttag,
6.00)
Figure 7.1
Some items in the “inchargeof” relation
T
between faculty and sub
ject numbers.
where the faculty member named
h
instructorname
i
is in charge of the subject with
number
h
subjectnum
i
. So
T
contains pairs like those shown in Figure
7.1
.
This is a surprisingly complicated relation: Meyer is in charge of subjects with
three numbers. Leighton is also in charge of subjects with two of these three
numbers—because the same subject, Mathematics for Computer Science, has two
numbers (6.042 and 18.062) and Meyer and Leighton are jointly inchargeof the
subject. Freeman is inchargeof even more subjects numbers (around 20), since
as Department Education Officer, he is in charge of whole blocks of special sub
ject numbers. Some subjects, like 6.844 and 6.00 have only one person incharge.
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Databases, Equivalence relation, Binary relation, Transitive relation, Partially ordered set, partial order, Partial Orders

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