SOLUTIONS TO HOMEWORK 5 MATH 150, FALL 09
Problem 1. Section 2.2/ Exercise 8. Suppose is a set of formulas such that for all sentences , either |= or |= . Assume that A |= . Show that for any , A |= i |= . For one of the directions, suppose that |= . Then
SOLUTIONS TO PRACTICE MIDTERM VERSION 2
(1) [20] True or false. Just write the word TRUE/T or FALSE/F. No explanation is necessary. (a) Let x1 = (1, 0, 1)T , x2 = (1, 1, 3)T , and x3 = (2, 1, 4)T . The set cfw_x1 , x2 , x3 spans R3 . FALSE (b) Let A, B R
SFcrrox) 23 PARrt
trACtaR\ 4,ATis^J
oF
kf F be-o- (.il,J.
A r-?*rnual
pcx)e Ftxl
$<u6r* ovr F
(n-n $"A p.lqprnroJs
(*) e F [^l
tlui),h
&ff". , s.ih Ti.crT qc,t;
BC^lh($.
o
rt
.+
\pe"
Iou-na,r-
21'"(t-t
e iRl,<l f"d?'ts o$er iR
as G_rJ(*+rJx'r\ 6 lRkf Joes
MATH 120B, HOMEWORK 2, DUE Wednesday January 22
If you nd the homework problems hard, you may start by doing some problems from
the book (especially the ones with solutions). Remember that the only way to learn math
is to practice. The more problems you d
MATH 120B, HOMEWORK 4, DUE Thursday February 6
If you nd the homework problems hard, you may start by doing some problems from
the book (especially the ones with solutions). Remember that the only way to learn math
is to practice. The more problems you do
MATH 120B, HOMEWORK 5, DUE Thursday February 13
If you nd the homework problems hard, you may start by doing some problems from
the book (especially the ones with solutions). Remember that the only way to learn math
is to practice. The more problems you d
MATH 120B, HOMEWORK 7, DUE Friday February 28
I strongly recommend you read section 26 on the book and the class notes before you
attempt the homework. Some of the problems are tricky, and you are welcome to discuss
them in the group study sessions. Altho
MATH 120B, HOMEWORK 6, DUE Tuesday February 25
I strongly recommend you read section 23 and 26 on the book and the class notes before you attempt the homework. Some of the problems are tricky, and you are welcome
to discuss them in the group study session
MATH 120B, HOMEWORK 8, DUE Friday March 7
I strongly recommend you read section 26 and 27 on the book and the class notes before you attempt the homework. Some of the problems are tricky, and you are welcome
to discuss them in the group study sessions. Al
‘ ﬂn§w€V‘Kgﬂ
Math 120B — Quiz 1
January 16, 2014
Name
Student ID number
1. How does one determine if a subset S of a ring R is a subring of R?
“I? ﬁnk gt} {1W {3 ii%
QM 5;- {3 Dismal amt/e?" ma H“; [9? £53 ‘57 0W
2. Find all the zero diViSOIS of Z12 (if
Math 120A Midterm
November 1, 2013
You have until 10:50 AM to complete the exam. No calculators, books, notes, or consultation with other members
of the class are permitted. Your exam should have 5 pages.
Unsupported or improperly supported answers will r
SAMPLE MIDTERM 1, MATH 120A
(1)
1. Write the denition of group.
2. Give an example of a group of order 2011.
3. Give an example of a non-abelian group with 12 elements.
(2) Let (G, ) and (H, ) be groups.
1. Dene what it means for G and H to be isomorphic.
Supplementary problem Week 4
1. Let p be a prime number, and consider the set Z = cfw_ 1, 2, . . . , p 1 p of nonzero residue classes modulo p. Since p is prime, these are the residue classes of integers relatively prime to p. Recall that these then form
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the worlds books discoverable online.
It has survived long enough for the copyright to expire and
Chrystal, S09
Math 192 Final Exam
Give complete answers to each of the following questions and email your answers, as a Word attachment, to Larry4@cox.net by 10 P.M. on Tuesday, June 9, 2009 1. Which statement in Psychology and Behavior of Effective Teach
SOLUTIONS TO HOMEWORK 1 MATH 150, FALL 09
Problem 1. Section 1.1/ Exercise 2 If is a formula, then either it is a sentence symbol or it is build by one of the formula building operations applied to other formulas. In the rst case its length is 1, and in t
SOLUTIONS TO HOMEWORK 2 MATH 150, FALL 09
Problem 1. Section 1.2/ Exercise 9 Let be a w. We show that () is tautologically equivalent to by induction on the complexity of . Base step: suppose that is a sentence symbol, say = A. Then = (A), and so () = (A)
SOLUTIONS TO HOMEWORK 3 MATH 150, FALL 09
Problem 1. Section 1.5/ Exercise 12 cfw_, , is not complete. To show that, we claim that for every with one sentence symbol A, and connective symbols among , , , either is a tautology, a contradiction, or A |= .
SOLUTIONS TO HOMEWORK 4 MATH 150, FALL 09
Problem 1. Section 2.1/ Exercise 1 Part Part Part Part Part Part (a): x(N x 0 < x) (b): x(N x (Ix I 0) (c): x(N x (x < 0) (d): x(N x (Ix (y (N y (y < x Iy ) Ix) (e): x(N x y (N y y < x) (f ): x(N x y (N y (y < x)
MATH 120B: PRACTICE PROBLEMS FOR THE
MIDTERM
There will be ve problems on the midterm. The following are sample
problems.
Problem 1. Let R be a ring. An element a R is called nilpotent
if for some positive integer n, an = 0. Show that R has no nonzero
nil
MATH 120B, SPRING 2011
PRACTICE PROBLEMS FOR THE FINAL
Problem 1. Show that x4 22x2 + 1 is irreducible over Q.
Problem 2. Solve the congruence:
(1) 36x 15(mod24)
(2) 15x 27(mod18)
Problem 3. Show that if F is a eld with characteristic p for some
prime p,
MATH 120B: SOLUTIONS TO THE PRACTICE
PROBLEMS FOR THE MIDTERM
There will be ve problems on the midterm. The following are sample
problems.
Problem 1. Let R be a ring. An element a R is called nilpotent
if for some positive integer n, an = 0. Show that R h