EECS 203: Homework 9 Solutions
Section 8.1
1. (E) 4
a) antisymmetric, transitive
b) reﬂexive, symmetric, transitive
c) reﬂexive, symmetric, transitive
d) reﬂexive, symmetric
2. (E) 10 Only the relation in part (a) is irreﬂexive.
3. (E) 20
An asymmetric relation
R
must also be antisymmetric because there are no
(
a,b
)
∈
R
such that
(
b,a
)
∈
R
.
This vacuously satisﬁes the deﬁnition of antisymmetric.
An antisymmetric relation may not be asymmetric. For example, consider a relation
R
on the set
{
a
}
such
that
R
=
{
(
a,a
)
}
. This is antisymmetric but not asymmetric.
4. (E) 24
a)
R

1
=
{
(
a,b
)

a > b
}
b)
R
=
{
(
a,b
)

a
≥
b
}
5. (M) 46ab
a) There are only two relations on a set with one element, and both are transitive.
b) There are 13 transitive relations on a set with two elements, which can easily be found by enumerating
them. Alternatively, the following reasoning can be used.
Observe that a relation on two elements is always transitive when it does not contain both
(
a,b
)
and
(
b,a
)
. There are
2
2

1 = 3
ways to select
(
a,b
)
and
(
b,a
)
without choosing both at once, and there
are
2
2
= 4
ways to select the selfloops
(
a,a
)
and
(
b,b
)
. This is
3
·
4 = 12
transitive relations that do
not contain both
(
a,b
)
and
(
b,a
)
. There is only one transitive relation that contains