ECON 4117/5111 Mathematical Economics

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1 Test 1 September 23, Suppose that p and q are logical statements. The exclusive or, denoted by p Y q, is true when only one of p and q is true. (a) Construct the truth table of p Y q. (b) Prove or disprove the following statement with a truth table: (p Y q), [(p _ q) ^ (p ^ q)]. 2. In each part below, the hypotheses are assumed to be true. Use the tautologies from the appendix to establish the conclusion. Indicate which tautology you are using to justify each step. Notice that c denotes a contradiction. Appendix Some Useful Tautologies (a) (p ^ q), ( p) _ ( q) (b) (p _ q), ( p) ^ ( q) (c) [8 x, p(x)], [9 x 3 p(x)] (d) [9x 3 p(x)], [8 x, p(x)] (e) (p ) q), (p ) p 1 ) p 2 ) )q 2 ) q 1 ) q) (f) (p ) q), [( q) ) ( p)] (g) (p ) q), [p ^ ( q) ) c] (h) p, ( p ) c) (i) [p ) (q _ r)], [(p ^ ( q)) ) r] (j) [(p _ q) ) r], [(p ) r) ^ (q ) r)] (k) [p ^ (p ) q)], q (l) (p ) q), [p ^ ( q)] (m) [(p _ q) ^ ( q)] ) p (n) [(p ) q) ^ (r ) s) ^ (p _ r)] ) (q _ s) (a) Hypotheses: p ) u, r ) v, p ) q, r ) s, (u ^ v) Conclusion: q _ s (b) Hypotheses: r ^ s ) c, t ) u, s _ t Conclusion: r _ u 3. Suppose that n is a nonzero rational number. Show that if x is an irrational number, then nx is an irrational number. 4. Prove that n is an even number if and only if n 2 is even. 5. Let A, B, and C be sets in a universal set U. Prove or disprove the following statement: If then B = C. A \ B = A \ C,

2 Test 2 October 7, Let S be the set of all people living in Thunder Bay. Suppose that two people a and b are related by R if a lives within one kilometre of b. Explain if R is (a) reflexive, (b) symmetric, (c) transitive, (d) circular, (e) asymmetric, (f) antisymmetric, (g) complete. Is R a partial order? 2. Let S be the set of all straight lines in the plane. Define the relation R on S as is parallel to. Prove or disprove: R is an equivalence relation on S. 3. Let A =[ /2, /2]. Define the function f : A! R by f(x) =sinx. Explain whether f is (a) everywhere defined, (b) injective (one-to-one), (c) surjective (onto). Is f a bijection? 4. Suppose that f : X! Y is an injective function. Let A X. Show that 5. Show that for all n 2 N, f 1 (f(a)) = A (2n 1) = n 2.

3 Test 3 October 28, Let X be an infinite set. For x, y 2 X, define ( 1 if x 6= y, (x, y) = 0 if x = y. Provide a brief explanation for each of the following questions: (a) Characterize the open ball at a point p 2 X for all r>0. (b) Suppose that a, b 2 X and let A = {a, b}. Does the set A has any limit point? (c) Is A open or closed? (d) Is A a connected set? (e) Is A compact? of {x n }. Provide a brief explanation for each of the following questions: (a) Is {x n } an increasing sequence? (b) Is {x n } a bounded sequence? If yes, what is the set of upper bound? (c) Does inf A exists in Q? If yes, what is it? (d) Is {x n } a Cauchy sequence in R? 4. Suppose that f is a continuous function which maps a metric space (X, X ) into a metric space (Y, Y ). Let {x n } be a sequence in X which converges to a point x. Show that the sequence {f(x n )} converges to f(x) iny. 5. Let S =[5, 6] 2 R. Show that a continuous function f : S! S have at least one fixed point. 2. Let (X, ) be a metric space and A is an open set in X. Show that the complement, A c, is closed. 3. Consider the natural order on the set of rational numbers Q. Defineasequence{x n } in Q as follows: x 1 = 3, x 2 = 3.1, x 3 = 3.14, x 4 = 3.141, x 5 = , x 6 = ,. That is, x n+1 includes one more digit from the number than x n. Note that /2 Q. Let A be the range He asked totally unfair questions and someone purposely broke my pencil.

4 Test 4 November 11, Use the axioms of a vector space to prove the following results: (a) For all 2 R, 0 = 0. (b) If x = 0, theneither = 0 or x = Suppose that a linear operator f on R 2 is represented by the matrix 1 2 A =, 3 4 relative to the standard basis. (a) Identify the kernel of f. (b) What is the nullity of f? (c) What is the rank of f? Explain. 3. Suppose that V is a vector space with dim V > 2. Let S = {x, y} be a linearly independent set of vectors in V. (a) Define span(s), the linear span of S. (b) Prove or disprove: S? is a subspace of V. 4. Let f be a linear functional on a vector space V of dimension n. (a) Define the hyperplane of f at the level c 2 R. (b) What is the name of the hyperplane when c = 0? (c) Show that the set in part (b) is a subspace of V. 5. Let V be a normed vector space. Show that (V, ) is a metric space, where for every x, y 2 V, (x, y) =kx yk. Appendix A: Axioms for Vector Spaces Let V be a vector space with vector addition and scalar multiplication. For all x, y, z 2 V and, 2 R, 1. x + y = y + x 2. (x + y)+z = x +(y + z) V 3 x + 0 = x 4. 9 x 2 V 3 x +( x) =0 5. ( )x = ( x) 6. 1x = x 7. (x + y) = x + y 8. ( + )x = x + x Appendix B: Axioms for Normed Vector Spaces For all x, y in a vector space V and 2 R, 1. kxk 0, k0k = 0, 2. k xk = kxk, 3. kx + yk applekxk + kyk. Appendix C: Axioms for Metric Spaces A metric space consists of a set X and a distance function : X 2! R, which maps a pair of of points in X to a real number. The metric satisfies the following axioms: For all x, y, z 2 X, 1. (x, y) 0, (x, x) = 0, 2. (x, y) = (y, x), 3. (x, y)+ (y, z) (x, z).

5 Test 5 November 25, Consider the rotation function f : R 2! R 2 given by f(x 1,x 2 )=(x 1 cos where 2 [0, 2 ). x 2 sin, x 1 sin + x 2 cos ), (a) Find the matrix representation of f relative to the standard basis. (b) Does the inverse function f 1 exist? Explain. (c) If f 1 exist, find the matrix representation relative to the standard basis. If not, find the kernel of f. 2. Suppose that a linear operator f is represented by the matrix 1 2 A = 0 3 relative to the standard basis. diagonal and the o -diagonal elements are all zero. P =(x 1 x 2 x n ) is the matrix formed by the normalized eigenvectors. 4. A function f : R 2! R 2 is defined by f(x) =(x 1 + p x 2, (x 1 x 2 ) 1/2 ). (a) Find the derivative Df(x). (b) Does the inverse function of f exist in the neighbourhood of x =(1, 1)? Explain. 5. Consider the system of equations x x 2 = 1, x 1 +2x 2 = 3, where x =(x 1,x 2 ) and 2 R. (a) Express the system in the form f(x, )=0. (b) Show that the system has a solution x = g( ) where g : R! R 2 in the neighbourhood of the point (x 1,x 2, )=( 1, 2, 1). (c) Find Dg(1). (a) Find the eigenvalues of f and the normalize eigenvectors. (b) Find the rank of f. (c) What is the definiteness of A? 3. Let V be an n-dimensional inner product space. (a) Define a symmetric operator f on V. (b) Suppose that A is the matrix representation of f relative to a basis. Prove the Spectral Theorem: A = P P T where = diag( 1, 2,..., n), that is, is an n n diagonal matrix with the i at the principal

6 Final Examination December 16, 2016 Time Allowed: 2 hours are not required for the exam. 1. Let A be a set in a vector space V. (a) Define conv(a), the convex hull of A. (b) Suppose that B is also a set in V. Prove that 2. Let V be a vector space. conv(a + B) = conv(a) + conv(b). (a) Define an a ne set M in V. (b) Define the translation of M by a vector x 2 V. (c) Show that any translation of M is an a ne set. 3. Let f be a linearly homogeneous function on a convex cone S. Show that if f is concave, then for all x, z 2 S, f(x + z) f(x)+f(z). 4. Suppose that f(x, y) = 2x 2 + xy y 2 +3x + y +1. (a) Find the stationary point(s) of f. (b) Explain in details if the stationary point(s) above is (are) maximum, minimum, or saddle point. 5. Consider the following constrained optimization problem: (a) Set up the Lagrangian function. (b) Find the value function in terms of. (c) Find the rate of change of the value function with respect to. 6. Suppose that a consumer s preferences are represented by an increasing and quasi-concave utility function C 2 function u = f(x), where x 2 R n + is the consumption bundle. The budget constraint is given by p T x = M, wherep 2 R n ++ is the price vector and M 2 R + income. (a) Set up the utility maximization problem as a equality constrained optimization problem. (b) Find the bordered Hessian of the consumer maximization problem. 7. An inequality-constrained optimization problem is given by max x,y x 2 y 2 subject to x 2 + y 2 apple 4, y 0, y x. (a) Derive the Kuhn-Tucker conditions by assuming that only the first constraint is binding. Determine whether a solution exists with this assumption. (b) Find the solution of the problem. where >0. max x,y xy subject to x + y =, c 2016 Lakehead University. All Rights Reserved.