Math 330, Section 3: Number Systems
Fall 2010
Marcin Mazur
Oce: LN 2228
Oce Phone: 7-6540
E-mail: [email protected]
Course web page: www.math.binghamton.edu/mazur/teach/33011.html
Oce Hours: M, F 10:50-11:45; T 1:20-2:20. Also by appointment.
Text
Solutions for Exam 2, April 3, 2014, Math 304-Linear Algebra
Problem 1: 6 points
(a) (3 times 1 point) Dene what is:
(a.i) the span of three vectors v1 , v2 , and v3 in Rn ;
(a.ii) an ordered basis of a subspace V of Rn ;
(a.iii) an ane subset of Rn .
(b)
Solutions to Exam II
Problem 1. a) State the denition of a surjective function. How is the inverse function of a function
f : A B dened and when does it exist?
b) Let f : N N be a function dened by f (1) = 1 and for n > 1, f (n) is the smallest prime divi
Homework
due on Monday, February 7
Read carefully sections 2.3 and 7.1 in the book.
Problem 1. a) Dene 2n for every natural number n using the induction axiom.
b) Prove that n < 2n for every natural number n
c) Let a, b be natural numbers. Prove that if a
Solutions to Exam I
Problem 1. a) State all 9 axioms about addition and multiplication (4 about +, 4 about and one
connecting + and .) (7 points)
b) Using only the axioms for + prove that if a + c = b + c then a = b. (6 points)
c) Using only the axioms pr
Math 304 Section 07
Quiz 1
Name:
*You need to show all your work to get credit.*
1. What is the rank of a matrix (1 points)
2. Find the solutions of the following system of linear equations (2 points)
x1 + x2 + x3 = 1
x1 x2 + 2x3 = 0
2x1
x3 = 1
3. Find a
Solutions for Exam 3, April 30, 2014, Math 304-Linear Algebra
Problem 1: 6 points = 8 times 0.75 points
Test your understanding by marking each one of the following eight sentences as true (T) or false (F).
Please write your answers to the left of the sen
Math 304 Section 07
Quiz 2
Name:
*You need to show all your work to get credit.*
1 (1 point) Dene what is a function F from a set V to another set W .
2 (2 points) Let A be the function from R3 to R3 given by the following
formula A(x1 , x2 , x3 ) = (x1 +
Math 304 Section 07
Quiz 3
Name:
*You need to show all your work to get credit.*
(a) compute the inverse of
1
1
1 1
(1 point)
1 0 0
(b) compute the inverse of a 1 1 0 (2 points)
0 1 1
(c) Write
0 1
1 1
as a product of elementary matrices. (2 points)
1
Math 304 Section 07
Quiz 4
Name:
*You need to show all your work to get credit.*
1
1
Given three vetors v1 = ,v2 =
0
0
whether the following vectors are in the span
1
0
,v3 =
0
1
of v1 , v2 , v3
express that vector as a linear combination
Math 304 Section 07
Quiz 5
Name:
*You need to show all your work to get credit.*
(a) (3 points) We consider the following two bases X = cfw_p1 (x), p2 (x) and
Y = cfw_q1 (x), q2 (x) for P1 , where p1 (x) = 1 x, p2 (x) = 1 + x, q1 (x) = 1 + 3x,
q2 (x) = 1
Math 304 Section 07
Quiz 6
Name:
*You need to show all your work to get credit.*
(a)(2 points) Compute the determinant of the following matrices
3 3 3
3 4 3
4 4 3
.
(b)(3 points = 6 0.5 points) Test your understanding by marking each
one of the following
Solutions for Exam 1, March 3, 2014, Math 304-Linear Algebra
Problem 1: 6 points= 6 times 1 point Dene what is:
(a) A lower triangular matrix L of size n n.
(b) A pivot position of a matrix A of size m n.
(c) A linear transformation (map) T from a vector
Yaowen Song
Homework 4
Project 5.3 Define the following sets:
A := cfw_3x : x N
B := cfw_3x + 21 : x N
C := cfw_x + 7 : x N
D := cfw_3x : x N andx > 7
E := cfw_x : x N
F := cfw_3x 21 : x N
G := cfw_x : x N andx > 7
Determine which of following set eq
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\begincfw_document
\begincfw_flushleft
Yaowen Song
\endcfw_flushleft
\begincfw_center
Homework 2
\endcfw_center
\textbfcfw_Proposition 2.21. There exists no integer x such that $0<x<1$.
\textbfcfw_Proof: x$\notin$\textbfcfw_N
6.
Discuss the causes of the Great Depression.
What policies did
President Herbert Hoover attempt in trying to combat the effects on America?
Great Depression took place during the 1930th, it was the greatest,
deepest, and most widespread depression in th
\documentclass[11]cfw_article
\begincfw_document
\begincfw_center
\begincfw_flushright
Yaowen Song
\endcfw_flushright
\textbfcfw_Homework 1
\endcfw_center
\textbfcfw_Proposition 1.7. If m is an integer, then 0+m = m and 1$\cdot$m = m.\
\textbfcfw_Proof:0+
Yaowen Song
Homework 2
Proposition 2.21. There exists no integer x such that 0 < x < 1.
Proof: xN(prop
/
2.20), x6=0, and -xN,
/
according to Axiom 2.1 iv, mZ
/
Proposition 2.24. For all k N, k 2 + 1 > k.
Proof: let p(k)=k 2 + 1 k
p(1) = 1 > 0
assume p(k)
1.Discuss the issue of containment during the Cold War. What was the
idea surrounding this policy? Who were the major players? Was it successful?
The Cold War begins at 1947, soon after the World War II; and it end at
1989, when Soviet Union collapsed. Fi
Yaowen Song
Homework 4
Project 5.3 Define the following sets:
A := cfw_3x : x N
B := cfw_3x + 21 : x N
C := cfw_x + 7 : x N
D := cfw_3x : x N andx > 7
E := cfw_x : x N
F := cfw_3x 21 : x N
G := cfw_x : x N andx > 7
Determine which of following set eq
9.
What was Watergate, and why did it lead to President Richard
Nixons resignation?
Watergate is a name of a building; it is the center of Domestic party. The
Watergate incident happened at 1972 and it was also an election year On
several occasions Nixon
Yaowen Song
Homework 1
Proposition 1.7. If m is an integer, then 0+m = m and 1m = m.
Proof :0+m=m+0 (Axiom 1.1, i), m+0=m (Axiom 1.2). 1m=m1 (Axiom 1.1,
iv), m1=m(Axiom 1.3).
Proposition 1.8 If m is an integer, then (-m)+m=0.
Proof :(-m)+m=m+(-m) (Axiom 1
2. PROPOSITIONAL EQUIVALENCES
33
2. Propositional Equivalences
2.1. Tautology/Contradiction/Contingency.
Definition 2.1.1. A tautology is a proposition that is always true.
Example 2.1.1. p p
Definition 2.1.2. A contradiction is a proposition that is alwa
Homework
due on Tuesday, February 15
Read carefully sections 2.4 and 4.1 in the book. Solve the following problems.
Problem 1. Write a detailed proof of the following result. You can use the results
proved in class.
Theorem. If A is a bounded above set of
Homework
due on Wednesday, February 16
Read carefully sections 4.2-4.5 in the book. Solve the following problems.
Problem 1. Let n be a natural number. Verify the identity:
an+1 + bn+1 = (a + b)(an + bn ) ab(an1 + bn1 ).
Use it to prove that if a, b are r
Homework
due on Wednesday, February 2
Read carefully sections 2.1 and 2.2 in the book.
Problem 1. Using only the results from Chapter 1, from section 2.1, and the
properties proved in class prove that:
a) If m < n and a < 0 then am > an.
b) If a > 0 and a
Proposition 1.17.
(i) 0 is divisible by every integer.
(ii) If m is an integer not equal to 0, then m is not divisible by 0.
[Homework]
1.2 First Consequences 11
Proof. (i) Given m Z, we have 0 = 0 m by Proposition 1.14, and so by defnition
0 is divisible
Propositon 1.20. For all m,n Z, (m)(n) = mn.
Proof. Let m,n Z. By Axiom 1.4,
m+ (m) = 0 and n+ (n) = 0.
Multplying both sides of the first equaton (on the right) by n and the second equaton
(on the left) by m gives, after applying Propositon 1.14 on the r
Propositon 1.13. Let x Z. If x has the property that there exists an integer m such
that m+x = m, then x = 0.
[Homework]
Proof. Assume x,m Z satsfy m+x = m. Now by Axiom 1.4, we know that there
exists an integer m such that m+ (m) = 0. We add this number
Math 330: Number Systems
Spring 2017
Section 04
CLASS TIME: Mondays, Wednesdays, and Fridays 1:10 2:10, Thursdays 11:40 1:05
CLASS ROOM: WH 100B
INSTRUCTOR: Russell Ricks
EMAIL: [email protected]
OFFICE: WH 202
OFFICE HOURS: Mondays 2:10 3:30, Thu