COT3100.01, Fall 2000
S. Lang
Solution Key to Assignment #3 (40 pts.)
Due: 10/24 in class
First, we define some terms and notations.
Definition.
Consider a binary relation
R
⊂
A
×
B
.
The
inverse
of
R
, denoted
R
–1
, is a binary
relation
⊂
B
×
A
such that
R
–1
= {(
b
,
a
)  (
a
,
b
)
∈
R
}, that is,
R
–1
contains pairs of elements which
have the reverse order as they are in relation
R
.
(In some text
R
–1
is called the
converse
of
R
,
denoted
R
c
.)
Definition.
Let
R
⊂
A
×
A
denote a binary relation.
The following relations defined over
A
are
called
closures
:
(a)
The
reflexive closure
of
R
is
r
(
R
) =
R
∪
{(
a
,
a
) 
a
∈
A
}.
(b) The
symmetric closure
of
R
is
s
(
R
) =
R
∪
R
–1
.
(c)
The
transitive closure
of
R
is
t
(
R
) =
R
∪
R
2
∪
R
3
∪
...
, where
R
2
=
R
R
,
R
3
=
R
2
R
, etc.,
where
denotes relation composition.
Thus, (
a
,
b
)
∈
t
(
R
)
⇔
(
a
,
b
)
∈
R
n
,
for some
n
≥
1
⇔
there exist
a
1
,
a
2
, …,
a
n
∈
A
,
a
n
=
b
, for some
n
≥
1, such that (
a
,
a
1
), (
a
1
,
a
2
), …, (
a
n

1
,
a
n
)
∈
R
, i.e., there exists a direct path of
n
edges connecting
a
to
b
in the digraph for the relation
R
.
It can easily be seen that the names of these closures are justified in that for any binary relation
R
,
r
(
R
) is reflexive,
s
(
R
) is symmetric, and
t
(
R
) is transitive.
In fact, the relation
r
(
R
) is the
smallest relation containing
R
and is also reflexive;
s
(
R
) is the smallest relation containing
R
and
is symmetric;
t
(
R
) is the smallest relation containing
R
and is transitive. Also, we can define the
composition of these closures, e.g.,
tr
(
R
) =
t
(
r
(
R
)),
rs
(
R
) =
r
(
s
(
R
)), etc.
The following figures show a binary relation
R
depicted by its digraph representation, and the
corresponding closures:
Now, answer the following questions neatly and concisely.
All proofs need to be justified by
using the appropriate definitions, theorems, and logical reasoning.
1.
(15 pts.) Let
A
denote an arbitrary nonempty set, and
R
and
T
denote arbitrary binary
relations defined over
A
.
Determine if each of the following statements is
true
or
false
, and
give a brief
explanation
of your answer.
In the cases of a false statement, use