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Given name
Math 2000 Quiz #2 (Fall 2011)
Time limit: 20 minutes
ID#
There are 40 total points on this test.
No books, notes, calculators, or collaboration.
+
(1) (15pts) Prove by induction that, for every n ∈ N , we have
n
X
(4k − 5) = 2n2 − 3n.
k=1
Proof by induction. Define P (n) to be the assertion
n
X
(4k − 5) = 2n2 − 3n.
k=1
(i) Base case. For n = 1, we have
n
X
k=1
(4k − 5) =
1
X
(4k − 5) = 4(1) − 5 = −1 = 2(12 ) − 3(1) = 2n2 − 3n.
k=1
So P (1) is true.
(ii) Induction step. Assume P (n − 1) is true (and n ≥ 2). Then
!
n
n−1
X
X
(4k − 5) =
(4k − 5) + (4n − 5)
k=1
k=1
= 2(n − 1) − 3(n − 1) + (4n − 5)
2
= 2(n − 2n + 1) − (3n − 3) + (4n − 5)
2
induction
hypothesis
= (2n2 − 4n + 2) − 3n + 3 + (4n − 5)
= 2n2 − 3n.
So P (n) is true.
(2) (10 pts) Indicate whether each of the following sets is countable (C) or uncountable (U).
(Circle the correct response for each set.) You do not need to justify your answers!
Score = # correct answers − # wrong answers. (Items left blank will be ignored.)
U
C
U
R
N×Q
C
Z
U
N×R
{ x ∈ R | x2 + 1 < 5 }
U
{ (x, y) ∈ R × R | x + y = 5 }
C
{ x ∈ R | x2 + 1 = 5 }
C
{ x ∈ R | x2 + 5 < 1 }
{ (x, y) ∈ R × Q | x + y = 5 }
C
U
{ (x, y) ∈ R × Q | x + y < 5 }
(3) (15 pts) Short answer. You do not need to show your work.
Assume D and M are sets.
(a) (3 pts) What does it mean to say that D has the same cardinality as M ?
There is a bijection from D to M .
(b) (3 pts) What does it mean to say that D is countably infinite?
D has the same cardinality as N+ .
Alternate answer: There is a bijection from D to N+ .
(c) (3 pts) What is the Pigeonhole Principle?
If #A = m and #B = n, and m > n, then there does not exist a
one-to-one function from A to B.
Alternate answer. If #A > #B, then there does not exist a one-to-one
function from A to B.
Alternate answer. Let B and A1 , A2 , . . . , An be finite sets. If B ⊂
A1 ∪ A2 ∪ . . . ∪ An , and n < #B, then #Ai ≥ 2, for some i.
Alternate answer. If a mail carrier has m letters to distribute among
n mailboxes, and m > n, then at least one of the mailboxes will have to
get more than one letter.
(d) (3 pts) What does it mean to say that D is countable?
D is either finite or countably infinite.
(e) (3 pts) For k ∈ N, what does it mean to say that #D = k?
There is a bijection from D to {1, 2, 3, . . . , k}.
Alternate answer: D has the same cardinality as {1, 2, 3, . . . , k}.