Econ 4111
Professor: John Nachbar
Spring 2012
Homework 1
Answers
1. Let denote ( or ( and ) ( or ) and ).
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and or ( and ) or ( or ) and
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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 subject. I try to convey the main line
of development w
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 metric, d : X X R. The interpretation is that d(a, b)
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 implies, and quantiers, such as there exists. To understand t
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 bounded. Theorem 7 is immediate if I can show (a) that RN is
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 bounded and a
subset of R, it has a least upper bound, say y
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: for any set O of open sets with the property that
C
O
OO
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 < x
is an upper bound). Since the sequence is weakly inc
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, b, c. Then the implied
choice function has C(B1 ) = c
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) The Slater condition is that there is a point interior to
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) = x1 + x2
s.t. g1 (x) = (x1 1)2 + x2 1 0
2
g2 (x) = x
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 rational then x =
r + (p), which is the sum of two rationals
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) and suppose
that xt x. I must show that x f 1 (C). Since
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 other than
x . Since N (x ) A O A, the result follows.
Si
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 ), hence
y f (A1 ) f (A2 ), as was to be shown.
(b) Conside