Numbers and Polynomials
Test 3
Answer FOUR questions. Be sure to give reasons for each step.
1. Where possible, in each case give (with justication) an example of a
nonempty set of real numbers that is bounded above and has:
(i) a greatest element;
(ii) a
Numbers & Polys
MAS3300
Prof. JLF King
3Apr2006
Exam-U
Note. This is an open brain, open (pristine) SigmonNotes
exam, calculator permitted. Please write each of the two essays on separate sheets of paper, using complete grammatical
English sentences. Use
Numbers & Polys
MAS3300
V3: Consider the Fibonacci numbers (fn ) den=
ned by f0 := 0, f1 := 1, and n Z : fn+1 =
fn + fn1 . Prove by induction that
Prof. JLF King
20Nov2006
Home-V
Note. Permitted: Brain, SigmonNotes, calculator, com-
:
puter, webpage; but
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MAS 3300 Numbers & Polynomials
Solution to Homework 1
1.4 Theorem. Suppose a, b, c, d R. Then
(a) If a + c = b + c, then a = b.
Proof. Suppose a, b, c R and a + c = b + c.
Then,
( a + c) + ( c) = ( b + c) + ( c)
(by (AIV),
a + (c + (c) = b + (c + (c)
(by
MAS 3300 - FALL 2011 QUIZ 1 - SOLUTION
4. [10 pts] In this problem you may only assume results up to and including 1.4(l).
Suppose a R. Prove that
if a = 0 then
(a)1 = (a1 ).
Proof: Suppose a R and a = 0. Then a = 0 otherwise, if a = 0 then
a + (a) = 0,
(
1.15 Theorem.
(a) If a > 0 and b > 0, then a + b > 0.
Idea: Use the fact that a, b > 0 together with OA and OTR to show that a + b > 0.
Proof : Suppose a, b, c, d R, a > 0, and b > 0. Then
a+b>0+b
a+b>b
(by (OA),
(by (AID),
Hence, since b > 0
a+b>0
(by (O
Numbers & Polys
MAS3300
Prof. JLF King
8Mar2006
Note. This is an open brain, open (pristine) SigmonNotes exam. Please write each solution on a separate sheet
of paper.
Please be sure to write expressions unambiguously e.g, the expression 1/a + b should be
Set theory Miscellany
Jonathan L.F. King
University of Florida, Gainesville FL 32611-2082, USA
[email protected]
Webpage http:/www.math.u.edu/squash/
1 May, 2006 (at 02:10 )
For m a natnum, the value P (m) is a natnum-index
pair. Write this pair as (mN
Numbers & Polys
MAS3300 3244
Exam-S
Prof. JLF King
7Nov2005
Total:
275pts
Ordinal:
Note. This is an open brain, open (pristine) SigmonNotes exam. Please write each solution on a
Please be sure to write exseparate sheet of paper.
pressions unambiguously e.
Numbers & Polys
MAS3300
Prof. JLF King
4Jun2008
A1:
A2: 1.4h: (P.2) If b R then [ 1] b = b.
A3: Prove the triangle ineq., Thm1.20g:(P.6)
If x, y R then |x| + |y | |x + y |.
40pts
Total:
Print
name
30pts
A5:
A1: Thm1.4f: (P.2) If c R then c 0 = 0.
55pts
A4
Numbers & Polys
MAS3300
Prof. JLF King
4Jun2008
A1:
45pts
A2:
Exam-A
45pts
A3:
50pts
A4:
50pts
A5:
70pts
Bonus:
10pts
Total:
260pts
Note. This is an open brain, open (pristine) SigmonNotes exam. Please write each solution on a separate sheet
of paper.
Ple
Numbers & Polys
MAS3300
Exam-B
Prof. JLF King
4Jun2008
Note. This is an open brain, open (pristine) SigmonNotes exam. Please write each solution on a separate sheet
of paper.
Please be sure to write expressions unambiguously e.g, the expression 1/a + b sh
Numbers & Polys
MAS3300
Prof. JLF King
4Jun2008
B1:
B1: Using that N is WOed, prove that there is no
negative natnum. (<Thm2.3)
Print
name
60pts
Total:
Every if must be matched by a then.
100pts
B4:
Notes exam. Please write each solution on a separate
she
Numbers & Polys
MAS3300
Exam-C
Prof. JLF King
4Jun2008
Note. This is an open brain, open (pristine) SigmonNotes exam. Please write each solution on a separate sheet
of paper.
Please be sure to write expressions unambiguously e.g, the expression 1/a + b sh
Numbers & Polys
MAS3300
Prof. JLF King
4Jun2008
C1:
80pts
C4:
80pts
C5:
80pts
Total:
Notes exam. Please write solutions for C3,C4,C5 on
separate sheets of paper.
Write expressions unambiguously e.g, 1/a + b should be bracketed either [1/a] + b or
1/[a + b
Numbers & Polys
MAS3300
g ( 2 )=
3
Prof. JLF King
4Jun2008
Home-D
Note. Please be sure to write expressions unambiguously e.g, the expression 1/a + b should be bracketed
either [1/a] + b or 1/[a + b]. Be careful with negative
signs!
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[email protected]
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1.15 b)
If a < b , then a > b
IDEA: Add b to both sides and use (OA).
Proof. Suppose a, b R and a < b.
Then,
a + (b) < b + (b)
a + (b) < 0
(by (AIV),
a + (b) + (a) < 0 + (a)
0 + (b) < 0 + (a)
Therefore,
(by (OA),
b < a
(by (OA),
(by (AA, AIV),
(by (AID),
1.15 Theorem.
Suppose a, b, c R.
(c) If a < b and c < 0, then ac > bc.
IDEA: The idea here is to use the fact that c < 0 to show that c > 0, and then to
simply use the OM. After that, we will use OA to switch the sides that the variables
are on.
Proof.
Su
1.15 Theorem.
Suppose a, b, c R.
(d)
(i) We must show that if a > 0 and b > 0, then it implies ab > 0.
(ii) We must show that if a > 0 and b < 0, then it implies ab < 0.
(iii) We must show that if a < 0 and b < 0, then it implies ab > 0.
IDEA: The rst thi
1.20 Theorem.
Suppose that a, b, c R. Then
(e) For c 0, |a| c i c a c.
IDEA: The idea here is to pay attention to the i of the statement to be proven.
We must show that the statement is true going both ways. Going one way, we start
with |a| c, we will use
1.20 f ) (i) |ab| = |a|b|
IDEA: Consider the square of both sides and use 1.20(c) and 1.17(a).
Proof. Suppose a, b R.
|a| 0, |b| 0, and |ab| 0 by 1.20a. Also |a|b| 0 by 1.15(d) and 1.4(f). By
1.20(c) we have
|ab|2 = (ab)2 = (ab)(ab) = aabb = a2 b2 = |a|2
1.20 (g) |a + b| |a| + |b|
IDEA: Use Theorem 1.20 (e), replacing a by a + b and c by |a| + |b|.
Proof: Suppose that a, b R. Then
|a| a |a|
(by (1.20 d), and
|b| b |b|
(by (1.20 d).
Then by adding the two equations, we have
(|a| + |b|) = (|a|) + (|b|) a +
Math 461
Abstract Algebra Part 1
Cumulative Review
Text: Contemporary Abstract Algebra by J. A. Gallian,
6th edition
This presentation by:
Jeanine Joni Pinkney
in partial fulfillment of requirements of Master of Arts in Mathematics
Education degree
Centra
Kenneth Bixgorin
MAS 4300
Professor Lebovitz
Fall, 2016
Assignment: how you got to this point and why you are pursuing a career in math education
-After teaching photography in the US Air Force during WWII, my father, Jack Bixgorin, attended Hunter
Colleg