RED
Name:
Fall 2016
Student Submitted Sample Questions for the final
Last Update: December 8, 2016
The questions are divided into three sections:
Truefalse, Multiple Choice, and Written
Answer. I will add questions to the end of each section, so the number of a question should
never change–you can refer to “T/F question 5” no matter when you download the file (as
long as it is after five questions have been put in).
1.
TrueFalse Questions
1. If we expand the expression (3
x
+
y
)
6
, the coefficient of the
x
5
y
term is 1458.
Proof.
True. The binomial coefficient of the
x
5
y
term is
(
6
1
)
, which equals 6. 6
*
3
5
*
1
1
= 1458.
2.
P
∧
(
P
=
⇒
Q
) is a tautology.
Proof.
False. Consider the truth table:
P
Q
P
=
⇒
Q
P
∧
(
P
=
⇒
Q
)
T
T
T
T
T
F
F
F
F
T
T
F
F
F
T
F
Therefore,
P
∧
(
P
=
⇒
Q
) is not a tautology.
3. The statements
P
=
⇒
Q
and ”For P, it is necessary that Q” have the same
meaning.
Proof.
True
4. If
A
×
B
=
∅
, then
A
=
∅
and
B
=
∅
.
Proof.
False. ”Then
A
=
∅
or
B
=
∅
” would make the statement correct.
5. The following statements are logically equivalent:
P
∧
(
Q
∨
R
)
≡
(
P
∧
Q
)
∨
(
P
∧
R
).
Proof.
These statements are logically equivalent and can be shown to be logically
equivalent through a truth table.
6. Let
A
be the set
A
=
{
1
,
3
,
7
,
20
,
400
,
5000
}
. The size of the
P
(
A
) = 36.
Proof.
False. The size of the power set is 2

A

, not

A

2
.
7. If
P
and
Q
are statements, then (
∼
P
) =
⇒
(
P
=
⇒
Q
) is a tautology.
Proof.
True.
Since (
P
=
⇒
Q
) is true except when
P
is true and
Q
is false,
this is the only case we have to check, since otherwise, the truth value of the ”if”
statement does not matter. In this case,
P
is true, making
∼
P
false. Since both
the ”if” and ”then” statements are false, this is a tautology, since it will always be
true, regardless of the truth values of
P
or
Q
.
8. Any element
x
∈
Q
can be written as
a
b
, where
a, b
∈
N
.
1