MATH 277, AUTUMN QUARTER 2014
M. MALLIARIS
Course Instructor: Maryanthe Malliaris
Eckhart 318, mem@math.uchicago.edu
College Fellow: Jonathan Rubin
Course times: Tuesdays and Thursdays, 12-1:20pm, Eckhart 206
Class website: http:/math.uchicago.edu/mem/277
MATH 277, AUTUMN 2014
HOMEWORK 1: COUNTING
DUE WEDNESDAY, OCTOBER 8 AT 5PM
(1) Suppose A0 , A1 , B0 , B1 are disjoint sets. Suppose that for i = 0, 1 there
exists an injection from Ai to Bi but there does not exist a surjection. Let
A = A0 A1 . Let B = B0
MATH 277, AUTUMN 2014
REVIEW SHEET FOR MINI-MIDTERM 1
General Information: The rst Mini-Midterm will start at the beginning of class on
Thursday, October 16. Here is some information about the structure of the exam. It will
be 50 minutes long and consist
MATH 277, AUTUMN 2014
HOMEWORK 4
DUE WEDNESDAY, NOVEMBER 5 AT 5PM
(1) Prove that automorphisms preserve denable sets.
[That is, if M is an L-structure, f an automorphism of M , (x0 , . . . , xn ) is an L-formula whose free
variables are among x0 , . . . ,
MATH 277, AUTUMN 2014
HOMEWORK 3: MODELS, SUBMODELS, AUTOMORPHISMS
DUE WEDNESDAY, OCTOBER 29 AT 5PM
(1) Let L = cfw_< and consider the model M1 = (Q; <), i.e. the rationals in
which < is interpreted to mean the usual linear order. Prove that for any
a, b
MATH 277, AUTUMN 2014
TEST YOUR UNDERSTANDING
(1) Fix a language L = cfw_R where R is a binary relation symbol. Suppose M , N are countable
L-structures. Let ai : i < , bi : i < enumerate the domains of M and N respectively.
Call g a partial isomorphism f
MATH 277, AUTUMN 2014
HOMEWORK 8
DUE WEDNESDAY, DECEMBER 3 AT 5PM
(1) Prove that: (i) An ultralter on a set X is principal i it contains some nite set. (ii) Any
ultralter on a nite set is principal. (iii) If D is a nonprincipal ultralter on an innite
set
MATH 277, AUTUMN 2014
REVIEW SHEET FOR MINI-MIDTERM 2
General Information: The second Mini-Midterm will start at the beginning of class
on Thursday, November 13. Once again, the exam will consist of three questions (this time:
one xed, then choose two out
MATH 277, AUTUMN 2014
HOMEWORK 5
DUE WEDNESDAY, NOVEMBER 12 AT 5PM
(1) Suppose L contains equality and a single binary relation R. For each of the following three
L-sentences, (i) translate it into informal English and (ii) give an example of a model in
w
MATH 277, AUTUMN 2014
HOMEWORK 7
DUE WEDNESDAY, NOVEMBER 26 AT 5PM
(1) Let D be a nonprincipal ultralter on . Let M = D Fpi where pi is the ith prime and
p denotes the algebraic closure of the eld with p elements, considered in the language
F
L = cfw_+, ,
MATH 277, AUTUMN 2014
HOMEWORK 6
DUE WEDNESDAY, NOVEMBER 19 AT 5PM
(1) Work in the language L = cfw_+, , 0, 1 where +, are binary function symbols and 0, 1
are constants. Let p be a prime or 0.
(a) For each such p, write down a set of rst-order sentences
MATH 277, AUTUMN 2014
FINAL EXAM REVIEW
General Information: The nal exam will be on Tuesday, December 9 from 10:30am12:30pm in our usual classroom. There will be four questions (two xed, and then choose
two out of three). There may be an optional challen