COT3100
Summer’2001
Recitation on relations (Solutions)
06/14/2001
1. Let
A
={1, 2, 3},
B
={1, 2, 3, 4}, and
R
1
={(1,2), (2,2), (3,3)} and
R
2
= {(1,2), (1, 3), (1,4)} are relations from
A
to
B
. Find
R
1
∪
R
2
,
R
1
∩
R
2
, R
1

R
2
,
R
2

R
1
.
R
1
∪
R
2
= {(1, 2), (2, 2), (1, 3), (1, 4), (3, 3)}
R
1
∩
R
2
= {(1, 2)}
R
1

R
2
= {(2, 2), (3, 3)}
R
2

R
1
= {(1, 3), (1, 4)}
2. Given the set
A
= {2, 3, 4, 8, 9, 12, 18}, define a relation
T
over
A
such that
T
=
{(
a
,
b
)
a
∈
A
and
b
∈
A
and
ab
is a square number, i.e.,
ab
=
c
2
for some integer
c
}.
Answer the following two questions.
a)
Use a directed graph to depict the relation
T
defined above.
b)
Determine if relation
T
satisfies each of the properties: irreflexive, symmetric, and
transitive.
Not irreflexive (it is reflexive)
symmetric
transitive
3. Suppose
R
and
S
are symmetric binary relations on a set
A.
Must the following
relations be symmetric? Give either proofs or counterexamples to justify your answers.
a)
R
∩
S
Proof.
Assume (
x
,
y
) is arbitrary element from
R
∩
S
, i. e. (
x
,
y
)
∈
R
∩
S
. We need to
prove that (
y
,
x
)
∈
R
∩
S
given
R
and
S
are symmetric. If (
x
,
y
)
∈
R
∩
S
, then (
x
,
y
)