COT3100
Spring’2001
Test #2 (Total 110 pts)
Solutions
1.
Consider the relation
R
= {(
a
,
c
), (
b
,
d
), (
c
,
c
), (
d
,
a
), (
d
,
b
)} on the set
A
={
a
,
b
,
c
,
d
}.
a)
(3pts) Find
r
(
R
), the reflexive closure of
R
.
r
(
R
) =
R
∪
{(
a
,
a
), (
b
,
b
), (
c
,
c
), (
d
,
d
)}=
= {(
a
,
c
), (
b
,
d
), (
c
,
c
), (
d
,
a
), (
d
,
b
), (
a
,
a
), (
b
,
b
), (
d
,
d
)}
b)
(3pts) Find
s
(
R
), the symmetric closure of
R
.
s
(
R
) =
R
∪
R
–1
= {(
a
,
c
), (
b
,
d
), (
c
,
c
), (
d
,
a
), (
d
,
b
), (
c
,
a
), (
a
,
d
)}
c)
(4pts) Find
t
(
R
), the transitive closure of
R
.
One way is to find powers of
R
R
2
= {(
a
,
c
), (
b
,
b
), (
b
,
a
), (
c
,
c
), (
d
,
c
), (
d
,
d
)};
R
3
= {(
a
,
c
), (
b
,
d
), (
b
,
c
), (
c
,
c
), (
d
,
a
), (
d
,
b
), (
d
,
c
)};
R
4
= {(
a
,
c
), (
b
,
b
), (
b
,
c
), (
c
,
c
), (
d
,
d
), (
d
,
c
)}
Or it is possible to take all pairs connected with a path:
t
(
R
) = {(
a
,
c
), (
b
,
d
), (
b
,
b
), (
b
,
a
), (
b
,
c
), (
c
,
c
), (
d
,
a
), (
d
,
b
), (
d
,
d
), (
d
,
c
)}
2.
(8pts) Suppose
R
is a relation on a set
A
. Prove or disprove, that if
R
is symmetric and
transitive, then
R
is reflexive.
The proposition is false and can be disproved by the following example:
A
={
x
,
y
,
z
},
R
={(
x
,
x
), (
x
,
y
), (
y
,
x
), (
y
,
y
)} is symmetric and transitive, but is not
reflexive, because (
z
,
z
)
∉
R
.
3. Let
R
and
S
be relations on
X
. Determine whether each statement is true or false. If the
statement is false, give a counterexample.