MAS 111
FOUNDATION OF MATHEMATICS
ASSIGNMENT 9
HINTS
1. The following two binary relations are defined on the set
A
=
{
0
,
1
,
2
,
3
}
.
For each relation, determine whether it is reflexive, symmetric, tran
sitive. Give a counterexample in each case in which the relation does
not satisfy one of the properties.
(a)
R
1
=
{
(1
,
2)
,
(2
,
1)
,
(1
,
3)
,
(3
,
1)
}
(b)
R
2
=
{
(0
,
0)
,
(0
,
1)
,
(0
,
3)
,
(1
,
1)
,
(1
,
0)
,
(2
,
3)
,
(3
,
3)
}
Hint
:
(a)
R
1
=
{
(1
,
2)
,
(2
,
1)
,
(1
,
3)
,
(3
,
1)
}
is symmetric, but not reflexive
(for example, (1
,
1) is not in
R
1
), not transitive (for example,
(1
,
2)
,
(2
,
1) are in
R
, but (1
,
1) is not in).
(b)
R
2
=
{
(0
,
0)
,
(0
,
1)
,
(0
,
3)
,
(1
,
1)
,
(1
,
0)
,
(2
,
3)
,
(3
,
3)
}
is transitive,
but not reflexive ((2
,
2) is not in
R
), not symmetric ((2
,
3) is in
R
,
but (3
,
2) is not in).
2. Let
R
=
{
(0
,
1)
,
(0
,
2)
,
(1
,
1)
,
(1
,
3)
,
(2
,
2)
,
(3
,
0)
}
. Find
R
t
, the transitive
closure of
R
.
Hint
: It is easy to use directed graph to find
R
t
.
R
t
=
{
(0
,
1)
,
(0
,
2)
,
(1
,
1)
,
(1
,
3)
,
(2
,
2)
,
(3
,
0)
,
(0
,
3)
,
(1
,
0)
,
(3
,
1)
,
(1
,
2)
,
(3
,
2)
}
.
3. Let
S
=
{
(0
,
0)
,
(0
,
3)
,
(1
,
0)
,
(1
,
2)
,
(2
,
0)
,
(3
,
2)
}
. Find
S
t
, the transitive
closure of
S
.
Hint
: Easy.
4. Let
A
=
{
21
,
23
,
24
,
29
,
30
,
31
,
36
,
37
,
40
,
41
,
42
,
43
,
45
}
and let the rela
tion
R
on
A
be defined by
x R y
⇔
x

y
is divisible by 3
.