F06/CS 191 :
Suggested Solutions to
TEST TWO
Discrete Structures I
11/2/2006
OPEN BOOK
Name_____________________
There are 5 problems with 20 points each.
1
Let
X
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.
Define the relation
R
on
X
as follows.
(
a
,
b
)
∈
R
if
a
mod
3 =
b
mod
3
for any
a
,
b
∈
X
(a)
Prove that
R
is an equivalent relation.
Proof.
We need to show that
R
is reflexive, symmetric and transitive.
(i)
Because
a
mod
3 =
a
mod
3 for any number
a
, we have (
a
,
a
)
∈
R
.
So,
R
is reflexive.
(ii)
If (
a
,
b
)
∈
R
,
then we have
a
mod
3 =
b
mod
3. This implies
b
mod
3 =
a
mod
3. Therefore, (
b
,
a
)
∈
R. So
R
is symmetric.
(iii)
If (
a
,
b
)
∈
R
and
(
b
,
c
)
∈
R
, then we have
a
mod
3 =
b
mod
3, and
b
mod
3 =
c
mod
3, which implies that
a
mod
3 =
c
mod
3. Therefore, we have (
a
,
c
)
∈
R
. The transitivity is also proved.
Therefore,
R
is an equivalence relation.
(b)
List the elements of [3], the equivalent class containing 3.
[3] = {3, 6, 9, 12, 15}
1
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2
Let
X
= {2, 3, 4, 5},
Y
=
{6, 9, 10, 12},
Z
= {2, 7, 11, 13} be three sets. We define two
relations.
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 Fall '06
 Shen
 Mathematical Induction, Inductive Reasoning, Equivalence relation, Transitive relation, Structural induction

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