MATH 220.501
Examination 2
October 23, 2003
NAME
SIGNATURE
This exam consists of 9 problems, numbered 1–9. For partial credit you must present your
work clearly and understandably.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 9 PROBLEMS ON
7 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1
2
3
4
5
6
7
8
9
Total
Points
Possible
12
5
5
14
14
17
5
14
14
100
Credit
NAME
1.
MATH 220
Examination 2
Page 2
[12 points] Consider the following statement:
For all u ∈ Z, if there is an integer m so that u = 4m + 3, then there are no integers
a and b so that u = a2 + b2 .
Write out the negation, converse, and contrapositive of this statement.
Negation:
Converse:
Contrapositive:
2.
[5 points] Negate the following statement:
∀x ∈ R, (∃y ∈ R so that sin(y) = x) if and only if −1 < x < 1.
3.
[5 points] Negate the Well-Ordering Principle.
October 23, 2003
NAME
4.
MATH 220
Examination 2
Page 3
[14 points] Using only the axioms of Z (A0–A11), prove:
For all a, b ∈ Z, −(a − b) = b − a.
5.
[14 points] Prove the following statement:
For all x ∈ Z, if x2 ≤ 10x, then 0 ≤ x ≤ 10.
(Hint: Prove the contrapositive.)
October 23, 2003
NAME
6.
MATH 220
Examination 2
Page 4
[17 points] Fill in the blank so that the following statement is true. Then prove the
statement.
For all x ∈ Z, x3 − 4x2 = 0 if and only if x =
.
October 23, 2003
NAME
MATH 220
Examination 2
Page 5
7.
[5 points] What is the coefficient of x100 y 103 in the polynomial (8y − 5x)203 ? (No need
to simplify your answer!)
8.
[14 points] Prove by induction:
∀n ∈ Z+ ,
1
1
1
n
+
+ ··· +
=
.
1·2 2·3
n(n + 1)
n+1
October 23, 2003
NAME
9.
MATH 220
Examination 2
Page 6
[14 points] Prove by induction: For all n ∈ Z+ , the number n3 − n is divisible by 3.
October 23, 2003
NAME
MATH 220
Examination 2
Page 7
Known Axioms and Propositions
Axioms of Z:
A0.
A1.
A2.
A3.
A4.
A5.
A6.
A7.
A8.
A9.
A10.
A11.
Closure of Z: Z is closed under + and ·.
Associtativity of +.
Commutativity of +.
0 is the additive identity element.
Every integer has an additive inverse.
Associativity of ·.
Commutativity of ·.
1 is the multiplicative identity element.
Distributive law. (Include a(b + c) = ab + ac and (b + c)a = ba + ca.)
Closure of Z+ .
Trichotomy law.
Well-ordering principle.
Let a, b, c ∈ Z.
P1.
P2.
P3.
P4.
P5.
P6.
P7.
P8.
Subtracting from both sides: a + b = a + c ⇒ b = c.
a · 0 = 0 · a = 0.
(−a)b = a(−b) = −(ab).
−(−a) = a.
(−a)(−b) = ab.
a(b − c) = ab − ac.
(−1)a = −a.
(−1)(−1) = 1.
Let a, b, c ∈ Z. Statements Q2–Q9 hold equally well if < is replaced by ≤ and > is replaced
by ≥.
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
Exactly one of the following holds a < b, b < a, or a = b.
(a > 0 ⇒ −a < 0) and (a < 0 ⇒ −a > 0).
If a > 0 and b > 0, then a + b > 0 and ab > 0.
If a > 0 and b < 0, then ab < 0.
If a < 0 and b < 0, then ab > 0.
If a < b and b < c, then a < c.
If a < b, then a + c < b + c.
If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc.
Other statements we have proved (in homework or in class).
R1.
R2.
R3.
R4.
R5.
∀a, b ∈ Z, −(a + b) = −a − b.
∀a, b ∈ Z, if ab = 0, then a = 0 or b = 0.
∀a, b, c ∈ Z, if ab = ac and a 6= 0, then b = c.
There is no integer x so that 0 < x < 1.
1 is the smallest element of Z+ .