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
squash@math.ufl.edu
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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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
McGILL UNIVERSITY
FACULTY OF SCIENCE
DEPARTMENT OF
MATHEMATICS AND STATISTICS
MATHEMATICS 189340B
ABSTRACT ALGEBRA AND
COMPUTING
Notes Distributed to Students
(Winter Term, 2000/2001)
W. G. Brown
November 30, 2001
Notes Distributed to Students in Mathemat