Econ 4111
Professor: John Nachbar
Spring 2012
Homework 2
Answers
1. (a) y f (A1 A2 ) i there is an x A1 A2 such that y = f (x). For such
x, either x A1 or x A2 , hence either y f (A1 ) or y f (A2 ), h
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 1
Answers
1. Let denote ( or ( and ) ( or ) and ).
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
T
T
F
F
F
F
and or ( and ) or ( or ) and
T
T
T
T
T
F
T
T
Econ 4111
Professor: John Nachbar
Spring 2012
Test 1
Answers
1. By LUB, there is a least upper bound x. By the denition of least upper
bound, for any > 0, there is a T such that xT > x (otherwise, x <
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 3
Answers
1. Let O be an open set containing x . Then there is an > 0 such that
N (x ) O. Since x is a limit point, N (x ) A contains a point oth
Econ 4111
Professor: John Nachbar
Spring 2016
Homework 4
Answers
1. (a) (xt ) = (0, 1/2, 2, 1/4, 4, 1/8, 8, . . . ). A = cfw_0, 1/2, 2, 1/4, 4, 1/8, 8, . . . . 0
is the only limit point of A but xt do
Econ 4111
Professor: John Nachbar
Spring 2016
Homework 3
Answers
1. (a) h : R R is defined by h(x) = 2x x2 .
(b) x h1 (C) iff g(f (x) C iff f (x) g 1 (C) iff x f 1 (g 1 (C), which
is to say x f 1 g 1
Econ 4111
Professor: John Nachbar
Spring 2017
Homework 2
Answers
1. (a) f 1 (B) is the disk, including the boundary, centered at the origin, of
radius 1.
(b) f 1 (B) is the half-space, not including t
Econ 4111
Professor: John Nachbar
Spring 2017
Homework 1
Answers
1. Let denote ( or ( and ) ( or ) and ).
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
T
T
F
F
F
F
and or ( and ) or ( or ) and
T
T
T
T
T
F
T
T
Due: 11:59PM, 9/20/17
Health Economics ECON 352.01
Prof. Grace J. Y. Johnson
Problem Set 1
1. Uncertainty enters into the maintenance and production of health in numerous ways,
including (a) Whether a
Econ 407 Homework 1 Suggested
Solutions
Erdem Yenerdag
September 19, 2017
Problem 1
a. The man-proposing deferred acceptance algorithm finds the man-optimal stable
matching. In the first stage, w2 mat
Econ 4111
Professor: John Nachbar
February 9, 2012
Set Theory
1
Overview
This is an informal introduction to Set Theory, which is somewhat ironic because
set theory is, by its nature, a highly formal
Econ 4111
Professor: John Nachbar
September 30, 2011
Metric Spaces
1
Metric Spaces Basics.
1.1
Metric spaces.
A metric space (X, d) consists of a set of points, X together with a distance function,
or
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 4
Answers
1. (a) Consider any closed set C. If f 1 (C) = then I am done, because is
closed. Otherwise, consider any sequence cfw_xt in f 1 (C) a
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 5
Answers
1. Let x be any element of S. Then a x and x b. By transitivity, a b.
2. (a) By contraposition. If p is rational and p + x = r is ratio
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 6
Answers
1. (a) The constraint set is lens-shaped and lies along the 45 degree line, with
one end at (0,0) and the other at (1,1).
(b)
maxx
f (x
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 7
Answers
1. (a) The solution is x = (1, 1).
(b) No, because
(c)
g(x ) = (0, 0).
f (x ) = (1, 1), so there is no such that
f (x ) = g(x ).
(d) Th
Econ 4111
Professor: John Nachbar
Spring 2012
Homework 8
Answers
1. Suppose a b, b S c, and c S a. Note that this is complete but not transitive.
Let B = cfw_B1 , B2 with B1 = cfw_a, b and B2 = cfw_a
Econ 4111
Professor: John Nachbar
December 14, 2010
Compactness
1
Introduction.
An important fact about metric spaces is that the following ve properties of a set
C are equivalent.
1. C is compact: fo
Econ 4111
Professor: John Nachbar
Spring 2012
Test 2
Answers
1. If C is compact and f : C R is continuous then f (C) is compact (by a
previous theorem), hence closed and bounded. Since f (C) is bounde
Econ 4111
Professor: John Nachbar
October 5, 2010
Compactness and Completeness in RN .
1
R is complete.
Theorem 7, the Heine-Borel theorem, states that a set in RN is compact i it is
closed and bounde
Econ 4111
Professor: John Nachbar
August 31, 2011
Logic and Proofs
1
Sentential Connectives and Tautologies
Formal logical statements are built up out of sentential connectives, such as and
and implie
Econ 407 in 2017: Problem Set 1. Due Sep 19th (Class time)
The first three problems are for your homework. You must support your answers
with convincing arguments. Additional problems are for your exe