A. Miller
M541
Exam 2 Answers
Fall 99
1. Problem-Points: 1-5 2-5 3-5 4-4 5-4 6-4 7-9 8-3 9-2 10-5 11-6 12-5 (a) f : X Y is one-to-one iff x, y X f (x) = f (y) x = y (b) f : X Y is onto iff y Y x X f (x) = y (c) g is the inverse of f iff g : Y X and x X y
Summary Part III, Ch. 2, Second part: Gulliver discovers the name
of the island is Laputa. As the capital city moved, they collected
petitions form the passing villages by ropes being sent to the land.
Their language is largely based on mathematics and mu
Exposure Java 2005
Name:
Exercises 8.1
Date:
Period:
1.
What is the single most important goal for any program?
2.
What does GUI stand for?
3.
What are the 3 cornerstones of OOP?
4.
What 2 things do objects store?
5.
What makes objects unique?
6.
Explain
1910- The week-end becomes a popular time in the U.S.
1911- The triangle Shirtwaist Fire
1912- The Titanic, the largest passenger ship in the world at the time sinks.
1912- Massachusetts becomes the first state to adopt a minimum wage
1914- The assassinat
OF MONARCHY AND HEREDITARY SUCCESSION
MANKIND being originally equals in the order of creation, the equality
could only be destroyed by some subsequent circumstance; the distinctions
of rich, and poor, may in a great measure be accounted for, and that wit
#
#
15.Is the sequence 3, 12, 36, . a geometric sequence? Explain.Type your answer
below.
NO! A geometric sequence goes from one term to the next by always multiplying
(or dividing) by the same value. So 1, 2, 4, 8, 16,. and 81, 27, 9, 3, 1,
1/3,. are geo
A. Miller
M541
Review for Exam 2
Fall 99
1. Dene f : X Y is one-to-one. Dene f : X Y is onto. Dene g is the inverse of f . 2. Prove f is 1-1 onto i f has an inverse. 3. Dene transposition, n-cycle, disjoint cycles, parity, sign, crossing pair, crossing nu
A. Miller
M542
An Example
Spring 2000
Theorem. There exists a eld F and , in some extension eld of F such that [F (, ) : F ] < but there is no F [, ] such that F (, ) = F (). This is similar to the example on page 354 of Gallians book. Let F = Z2 (s, t) w
A. Miller
M541
Exam 1 Answers
Fall 99
1. (4points) Suppose p, n, m are integers. (a) n divides m iff k Z m = kn. (b) p is a prime number iff p 2 and there exists no k Z with 1 < k < n and k divides p. (c) gcd(n, m) = d iff d is the largest integer which d
A. Miller
M542
Final Exam
Spring 2000
The Final Exam is in our usual classroom (B203 Van Vleck) at 7:25pm on Saturday May 13. It consists of approximately six proofs from the material below which I will write on the blackboard. A copy of this document wil
A. Miller
M542
Galois Theory
Spring 2000
For the material on Galois theory we will be assuming that the elds all have
characteristic zero. When we get to solvability by radicals we will assume that all
elds are subelds of the complex numbers C.
1
Review
T
A. Miller
M340
April 97
edited Jan 2000 for M542
Vector Spaces
A vector space, V , is a set with two operations, vector addition (written
u + v) and scalar multiplication (written av). The elements of V will be
denoted using u, v, w, etc. The formula u V
A. Miller
M542
Spring 2000
1
Linear Transformations
In this section we consider only nite dimensional vector spaces V or W over an
arbitrary eld F.
Theorem 1.1 Every linear transformation L : Fn Fm is determined by an m n
matrix A:
L(X) = AX
for every X F
A. Miller
M542
Review for midterm
Spring 2000
1. Lemma. Suppose G be a nite abelian group such that |G| = mn where m and n are relatively prime. Let H = cfw_x G : xn = e and K = cfw_x G : xm = e. Then H and K are subgroups of G and G = H K H K. 2. Lemma.