Economics 204
Fall 2012
Problem Set 1 Suggested Solutions
1. Use induction to prove the following statements.
(a) The equality
(b) The inequality
n
3
i=1 i =
n
1
i=1 i
(
n
i=1
2
i) holds for all n N;
n holds for all n N;
(c) The inequality (1 + x)n 1 + nx
Economics 204 Lecture 11Monday, August 10, 2009 Revised 8/10/09, Revisions indicated by * and Sticky Notes Sections 4.1-4.3, Unified Treatment Definition 1 Let f : I R, where I R is an open interval. f is differentiable at x I if f (x + h) - f (x) =a lim
Economics 204 Lecture Notes on Measure and Probability Theory This is a slightly updated version of the Lecture Notes used in 204 in the summer of 2002. The measure-theoretic foundations for probability theory are assumed in courses in econometrics a
Economics 204 Lecture 11Monday, August 10, 2009 Revised 8/10/09, Revisions indicated by * and Sticky Notes Sections 4.1-4.3, Unified Treatment Definition 1 Let f : I R, where I R is an open interval. f is differentiable at x I if f (x + h) - f (x) =a lim
Economics 204 Summer/Fall 2008 Lecture 1Monday July 27, 2009 Bob Anderson Vladimir Asriyan Hui Zheng Website: http:/emlab.berkeley.edu/users/anderson/Econ204 /204index.html Lectures will often run past 3:00, don't schedule things before 3:30. Sections: 9-
University of California, Berkeley Department of Economics Econ 204 Mathematical Tools for Economists Summer/Fall 2009 Revised 7/24/09, reordering the material in Lectures 11 and 12 (see * below). Instructors Bob Anderson 583 Evans [email protected]
Economics 204 Summer/Fall 2008 Lecture 1Monday July 27, 2009 Bob Anderson Vladimir Asriyan Hui Zheng Website: http:/emlab.berkeley.edu/users/anderson/Econ204 /204index.html Lectures will often run past 3:00, don't schedule things before 3:30. Sections: 9-
Economics 204 Lecture 2, July 28, 2009 Section 1.4, Cardinality (Cont.) Theorem 1 (Cantor) 2N , the set of all subsets of N, is not countable. Proof: Suppose 2N is countable. Then there is a bijection f : N 2N . Let Am = f(m). We create an infinite matrix
Economics 204 Lecture 3Wednesday, July 29, 2009 Revised 7/29/09, Revisions Indicated by * and Sticky Notes Section 2.1, Metric Spaces and Normed Spaces Generalization of distance notion in Rn Definition 1 A metric space is a pair (X, d), where X is a set
Economics 204 Lecture 4Thursday, July 30, 2009 Revised 7/31/09, Revisions Indicated by * and Sticky Notes Section 2.4, Open and Closed Sets Definition 1 Let (X, d) be a metric space. A set A X is open if xA>0B (x) A A set C X is closed if X \ C is open. E
Economics 204 Lecture 5Friday, July 31, 2009 Section 2.6 (Continued) Properties of Real Functions
Theorem 1 (6.23, Extreme Value Theorem) Let f be a continuous real-valued function on [a, b]. Then f assumes its minimum and maximum on [a, b]. In particular
Economics 204 Lecture 6Monday, August 3, 2009 Revised 8/4/09, Revisions indicated by * and Sticky Notes Section 2.8, Compactness Definition 1 A collection of sets U = cfw_U : in a metric space (X, d) is an open cover of A if U is open for all and U A ( m
Economics 204 Lecture 7Tuesday, August 4, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Note: In this set of lecture notes, A refers to the closure of A. Section 2.9, Connected Sets Definition 1 Two sets A, B in a metric space are separat
Economics 204 Lecture 8Wednesday, August 5, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Chapter 3, Linear Algebra Section 3.1, Bases Definition 1 Let X be a vector space over a field F . A linear combination of x1, . . . , xn is a vecto
Economics 204 Lecture 9Thursday, August 6, 2009 Revised 8/6/09, revisions indicated by * and Sticky Notes Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Definition 1 Given a vector space X and a vector subspace W of X, define an equ
Economics 204 Lecture 10Friday, August 7, 2009 Revised 8/8/09, Revisions indicated by * and Sticky Notes Diagonalization of Symmetric Real Matrices (from Handout): 1 if i = j 0 if i = j A basis V = cfw_v1 , . . . , vn of Rn is orthonormal if vi vj = ij .
Economics 204 Lecture 2, July 28, 2009 Section 1.4, Cardinality (Cont.) Theorem 1 (Cantor) 2N , the set of all subsets of N, is not countable. Proof: Suppose 2N is countable. Then there is a bijection f : N 2N . Let Am = f(m). We create an infinite matrix
Economics 204 Lecture 3Wednesday, July 29, 2009 Section 2.1, Metric Spaces and Normed Spaces Generalization of distance notion in Rn Definition 1 A metric space is a pair (X, d), where X is a set and d : X X R+ , satisfying 1. x,yX d(x, y) 0, d(x, y) = 0
Economics 204 Lecture 3Wednesday, July 29, 2009 Revised 7/29/09, Revisions Indicated by * and Sticky Notes Section 2.1, Metric Spaces and Normed Spaces Generalization of distance notion in Rn Definition 1 A metric space is a pair (X, d), where X is a set
Economics 204 Lecture 11Monday, August 10, 2009 Sections 4.1-4.3, Unified Treatment
Definition 1 Let f : I R, where I R is an open interval. f is differentiable at x I if f(x + h) - f(x) =a h0 h lim for some a R. This is equivalent to f(x + h) - (f(x) + a
Economics 204 Lecture 10Friday, August 7, 2009 Revised 8/8/09, Revisions indicated by * and Sticky Notes Diagonalization of Symmetric Real Matrices (from Handout): 1 if i = j 0 if i = j A basis V = cfw_v1 , . . . , vn of Rn is orthonormal if vi vj = ij .
Economics 204 Lecture 10Friday, August 7, 2009
Diagonalization of Symmetric Real Matrices (from Handout): Definition 1 Let ij =
1 if i = j 0 if i = j
A basis V = cfw_v1 , . . . , vn of Rn is orthonormal if vi vj = ij . In other words, each basis elemen
Economics 204 Lecture 9Thursday, August 6, 2009 Revised 8/6/09, revisions indicated by * and Sticky Notes Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Definition 1 Given a vector space X and a vector subspace W of X, define an equ
Economics 204 Lecture 9Thursday, August 6, 2009 Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Definition 1 Given a vector space X and a vector subspace W of X, define an equivalence relation by x y x-y W Form a new vector space X/W
Economics 204 Lecture 8Wednesday, August 5, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Chapter 3, Linear Algebra Section 3.1, Bases Definition 1 Let X be a vector space over a field F . A linear combination of x1, . . . , xn is a vecto
Economics 204 Lecture 8Wednesday, August 5, 2009 Chapter 3, Linear Algebra Section 3.1, Bases
Definition 1 Let X be a vector space over a field F . A linear combination of x1, . . . , xn is a vector of the form
n
y=
i=1
i xi where 1 , . . . , n F
i is the
Economics 204 Lecture 7Tuesday, August 4, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Note: In this set of lecture notes, A refers to the closure of A. Section 2.9, Connected Sets Definition 1 Two sets A, B in a metric space are separat
Economics 204 Lecture 7Tuesday, August 4, 2009 Note: In this set of lecture notes, A refers to the closure of A. Section 2.9, Connected Sets Definition 1 Two sets A, B in a metric space are separated if AB = AB = A set in a metric space is connected if it
Economics 204 Lecture 6Monday, August 3, 2009 Revised 8/4/09, Revisions indicated by * and Sticky Notes Section 2.8, Compactness Definition 1 A collection of sets U = cfw_U : in a metric space (X, d) is an open cover of A if U is open for all and U A ( m
Economics 204 Lecture 6Monday, August 3, 2009 Section 2.8, Compactness Definition 1 A collection of sets U = cfw_U : in a metric space (X, d) is an open cover of A if U is open for all and U A ( may be finite, countably infinite, or uncountable.) A set A