Math 100
Practice Final
Martin H. Weissman
On Thursday, December 13, there will be a threehour ﬁnal exam. The ﬁnal will be very
similar, in format and diﬃculty, to the series of questions given below.
Question:
1
2
3
4
5
6
7
Total
Points:
8
2
2
3
3
3
3
24
Score:
1. The following questions are worth 1 point each. No partial credit will be given.
(a) (1 point) Which of the following sentences are true (the ﬁrst, the second, both, or
neither)?
1.
∀
x
∈
Z
,
∃
y
∈
R
,
such that (
x
+
y > x
).
2.
∀
y
∈
R
,
∃
x
∈
Z
,
(
x
+
y > x
).
(a)
(b) (1 point) Let
A
=
{
1
,
3
,
5
,
7
,
9
}
. Let
B
=
{
1
,
4
,
7
,
10
}
. Let
C
=
{
1
,
5
,
6
,
9
}
. Describe
the set
A
∩
(
B
∪
C
). (An answer should look like “
{
2
,
3
,
5
}
”, though this is not the
correct answer).
(b)
(c) (1 point) The following deﬁnes a sequence
a
n
recursively:
•
Let
a
0
= 2. Let
a
1
= 3.
•
If
n
≥
2, and
n
∈
N
, then let
a
n
= 2
a
n

1

a
n

2
.
What is