Test 1 October 1, 2010 1. Construct a truth table for the following statement: [p (p q)] q. 2. A prime number is a natural number that is divisible by 1 and itself only. Let P be the set of all prime numbers and E be the set of even numbers. Consider the statement All prime numbers are odd numbers. (a) Write the statement in logic and set symbols using P and E. (b) Write the negation of the statement in plain English. (c) Prove or disprove the statement. 3. In each part below, the hypotheses are assumed to be true. Use tautologies from Figure 1 to establish the conclusion. Indicate which tautology you are using to justify each step. Figure 1: List of Tautologies for Question 3 (b) Show that the partition above induces an equivalence relation on S. 5. Let A and B be sets. Prove that A =(A B) (A \ B). (a) Hypotheses: r, t, (r s) t Conclusion: s (b) Hypotheses: s p, s r, q r Conclusion: p q 4. Let S be the set of all days in a year. Let A 1 = the set of all Sundays in S, A 2 = the set of all Mondays in S,. A 7 = the set of all Saturdays in S. (a) Show that the sets A 1,A 2,...,A 7 forms a partition of S. How am I supposed to think about consequences before they happen?
Test 2 October 15, 2010 and the left-side pages for your rough work. Do not forget 1. Let the set S = {1, 2, 3, 4, 5, 6}. Define an order on S as 5 3 1 4 2 6. Find the following: 3. 0 V x + 0 = x 4. x V x +( x) =0 5. (αβ)x = α(βx) 6. 1x = x 7. α(x + y) =αx + αy 8. (α + β)x = αx + βx (a) the interval [4, 3), (b) the upper contour set (1), (c) the best element of S, (d) inf S. 2. Let (X, ρ) be a metric space. Prove that a set A X is open if and only if A c is closed. 3. Let (X, ρ) be a metric space. Suppose that A and B are separated subsets of X. Prove or disprove: Ā B c. 4. Let {x n } be a sequence in a metric space (X, ρ). Suppose that x n converges to a point x. Show that {x n } is a Cauchy sequence. 5. Use the axioms for vector spaces in the Appendix to show that for all x, y, z V, (a) 0x = 0. (b) y + z = x + y implies that z = x. Appendix Axioms for Vector Spaces Let V be a vector space. For all x, y, z V and α, β R, there exists vectors x + y, y + z V such that 1. x + y = y + x 2. (x + y)+z = x +(y + z)
Test 3 October 29, 2010 and the left-side pages for your rough work. Do not forget (d) What is the hypograph of f? 1. Let V be a vector space. (a) Define a subspace S V. (b) Show that the zero vector, 0 S. (c) Show that for all x S, x S. (d) Is R n + a subspace of R n? Explain. 2. Consider the vector space R 3. (a) Show that the set of vectors B = {(2, 1, 0), (0, 2, 1), (0, 0, 1)} forms a basis for R 3. (b) Let x =(1, 2, 3). Find the coordinates of x with respect to the basis B. 3. Prove or disprove: (a) The union of two convex sets is convex. (b) The intersection of two convex sets is convex. 4. Let S be a subspace of a vector space V. (a) Show that S is a cone. (b) Is S a convex cone? Explain. 5. Define a function f : X R as 1 if x A, f(x) = 0 if x/ A, where A X. (a) Is f one-to-one? (b) Is f onto? (c) What is the graph of f?
Test 4 Novermber 12, 2010 1. Let S =[a, b] be a closed interval in R. Show that a continuous function f : S S have at least one fixed point. 2. Let f : V W be a linear transformation from vector space V into vector space W. (a) Define the rank of f. (b) State the dimension theorem. (c) Show that the range of f is a subspace of W. 3. Let B r (p) = {x X : ρ(x, p) r} be a closed ball centred at p with radius r in a metric space X. Let f be a continuous functional on X. Show that f( B r (p)) is bounded. Dear Math: I am not a therapist. Solve your own problems. 4. Let V be an inner product space. (a) State the Cauchy-Schwarz Inequality. (b) Define the angle between to vectors x, y V by cos θ = xt y xy. Prove that 1 cos θ 1. 5. Let f : V W be a linear transformation. Let S V be a convex set. Show that f(s) is convex.
Test 5 Novermber 26, 2010 1. Let A and B be n n square matrices. Prove that if A and B are invertible, then (AB) 1 = B 1 A 1. 2. Suppose that the matrix representation of a linear operator f : R 2 R 2 with respect to the standard basis is given by 0 1 A = 1 0 (a) Find the eigenvalues and the normalized eigenvectors. (b) What is the definiteness of f? Explain. 3. Let X be an n k matrix where k n and the columns of X are linearly independent. Show that X T X is positive definite. 4. Suppose that f : S R is a linearly homogeneous function. Show that f is also homothetic. 5. The determinant of a square matrix A is given by A = n ( 1) i+j a ij A ij, j=1 where A ij is the minor of the i-jth element. Show that if B is obtained from A by multiplying a row of A by a scalar α, then B = α A.
Final Examination December 17, 2010 Time Allowed: 2 hours 1. Find the directional derivative of the function f(x 1,x 2 )= 2x 2 1 + x 1 x 2 2x 2 2 3x 1 3x 2 at the point (1, 0) in the direction of (1, 1). 2. Suppose that f(x) = 1 4 x4 + 1 2 ax2 + bx, where a and b are parameters. (a) Find the set of critical points of f. (b) Apply the implicit function theorem to the necessary condition in part (a) to find the rate of change of the critical point x with respect to a and b. (c) Find the set of points (a, b) such that the implicit function theorem fails to apply. 3. Suppose that f : R n + R is a linearly homogeneous C 2 function. Prove the Euler Theorem: For all x R n ++, (a) f(x) = f(x) T x, (b) 2 f(x)x = 0. 4. Suppose a consumer wants to maximize the following function with respect to consumption c t and c t+1 : V t = U(c t )+βu(c t+1 ) (1) subject to the intertemporal budget constraint c t + c t+1 1+r = y t + y t+1 1+r, (2) where U is the instantaneous utility function, β is a discount factor, y t and y t+1 are incomes in periods t and t + 1 respectively, and r is the interest rate. (a) By writing (1) as f(x, θ) =U(c t )+βu(c t+1 ) V t with x = c t+1 and θ = c t, apply the implicit function theorem to find the marginal rate of time preference, dc t+1 /dc t. (b) Find the slope dc t+1 /dc t of the intertemporal budget constraint (2) as well. (c) Combine your results to get the Euler equation: βu (c t+1 ) U (1 + r) =1. (c t ) 5. The quadratic functional form f : R n + R is defined as f(x) =α 0 + α T x + 1 2 xt Ax, where α 0 R,α R n and A = A T is an n n symmetric matrix. (a) Find the gradient of f. (b) Find the Hessian of f. (c) Under what condition that f is a concave function? 6. Suppose that a consumer s preference structure can be represented by an increasing and quasi-concave C 2 function f : R n + R given by u = f(x). The consumer faces budget constraint p T x = M where p R n ++ is the market price of the bundle x and M is income. (a) Set up the consumer s optimization problem. (b) What is the Lagrangian function? (c) State the necessary conditions for optimization. (d) Find the bordered Hessian.
7. Solve the optimization problem max x,y x 1/3 y 2/3 subject to x +2y 10, x 0, y 0, using the Kuhn-Tucker Theorem. Of course they can do constrained optimization. 2