THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2012 Fundamental Concepts of Mathematics
Tutorial 3
1. Let
n
be an integer. Prove the following statements.
(a) If
n
is even, then 3
n
+
1 is odd.
(b)
n
2
+
n
+
1 is odd.
(c) If
n
is not a multiple of 5, then
n
4
-
1 is a multiple of 5.
2. Solve the following equations and inequalities.
Make sure that you understand the logical
relationships between the steps.
(a)
x
3
+
x
2
+
x
+
1
=
0 where
x
∈
R
(b)
√
x
+
1
+
4
=
2
x
where
x
∈
R
(c)
x
1
x
where
x
∈
R
{
0
}
3. Let
A,B,C
and
D
be sets. Prove the following statements.
(a) If
A
⊆
B
and
B
⊆
C
, then
A
⊆
C
.
(b) If
A
⊆
C
and
B
⊆
D
, then
(
A
×
B
)
⊆
(
C
×
D
)
.
(c)
(
A
∪
B
)
C
=
(
A
C
)
∪
(
B
C
)
.
4. Prove the following statements.
(a) Let
n
be an integer. If 2
n
2
+
1 is not a multiple of 3, then
n
is a multiple of 3.
(b) Let
a
be a nonnegative real number. If
a
<
x
for any positive real number
x
, then
a
=
0.
5. Explain why the following arguments are not correct.
(a) Solve the equation
x
2
-
1
=
0.
‘Solution’:
x
2
-
1
=
0
⇒
(
x
-
1
)(
x
+
1
)
=
0
⇒
x
-
1
=
0 or
x
+
1
=
0
⇒
x
=
1 or
x
=
-
1
So the solutions are
x
=
±
1.
(b) Prove that if
n
is an even number, then 1
-
n
is an odd number.
‘Solution’:
If
n
is even
⇒
n
=
2
k
,
k
∈
Z
⇒
1
-
n
=
1
-
2
k
=
2
(
-
k
)
+
1 is odd.
1
