Test 1 September 30, 2005 Read Me: Please write your answers on the answer book provided. Use the rightside pages for formal answers and the left-side pages for your rough work. Do not forget to put your name on the front page. 1. Consider the set S of all statements. Let a relation on S be implies ( ). For example, if p, q S, then p q means that p implies q. Using a truth table, show that this relation is transitive. 2. Classify the following statements as (i) logically true, (ii) a contradiction, or (iii) neither. (a) (p q) (q p) (b) (p q) (q p) (c) (p q) (p q) (d) (p q) (q r) (p r) (e) [(x A) (x B)] (x A B) 3. A prime number is a natural number that is divisible by 1 and itself only. Let P be the set of all prime numbers. Consider the statement If n is a prime number, then 2n + 1 is also a prime number. (a) Write the statement in logic and set symbols. (b) Write the contrapositive of the statement in plain English. (c) Prove or disprove the statement.
Figure 1: List of Tautologies for Question 4 4. 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. (a) Hypotheses: r s, r t, t u, v s Conclusion: v u (b) Hypotheses: r s, t u, s t Conclusion: r u 5. Let S and T be subsets of a universal set X. Prove that S T c = S \ T. (Note: any Venn diagram does not constitute a proof.) 2
Test 2 October 14, 2005 Read Me: Please write your answers on the answer book provided. Use the rightside pages for formal answers and the left-side pages for your rough work. Do not forget to put your name on the front page. 1. Let A = {2, 3, 4, 5, 6, 12, 15, 20}. Define a relation R on A as is a proper multiple of. (a) What is the domain of R? (b) What is the range of R? (c) Determine whether R is i. reflexive ii. transitive iii. symmetric iv. antisymmetric v. asymmetric vi. complete 2. Let be a weak order on a set X. (a) Define the lower contour set of x X. (b) Prove that if x y, then (y) (x). 3. Consider the Euclidean metric on R. Let A = ( 1, 1) (1, 2), that is, A is the union of two intervals. Answer the following questions with explanations. (a) What is the diameter of A? (b) Is A open, closed, or neither? (c) Is the point x = 1 a limit point of A? (d) What is A?
(e) Is A connected? 4. Let X be an infinite set. For all x, y X, define { 1 if x y, ρ(x, y) = 0 if x = y. Show that (X, ρ) is a metric space. 5. Using the axioms listed in the Appendix, prove that for every vector x in a vector space, (a) 0x = 0, (b) ( α)x = (αx). Indicate which axioms you use in each step of the proof. 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) 3. 0 V x + 0 = x 4. x V, x x + ( x) = 0 5. (αβ)x = α(βx) 6. 1x = x 7. α(x + y) = αx + βy 8. (α + β)x = αx + βy 2
Test 3 October 28, 2005 Read Me: Please write your answers on the answer book provided. Use the right-side pages for formal answers and the left-side pages for your rough work. Do not forget to put your name on the front page. 1. Consider the vectors a = (1, 1), b = ( 1, 1), c = (0, 1) in V = R 2. (a) Is B = {a, b, c} linearly independent? (b) Is B a basis of of V? Explain. (c) What is the linear hull of B? (d) On a diagram draw conv B. (e) Is conv B a convex cone? Explain. 4. (a) Let (X, X ) and (Y, Y ) be two ordered sets. Define a strictly decreasing function f : X Y. (b) Suppose u : R n + R is a utility function and g : R R is a strictly decreasing function. Is the transformation g u a utility function representing the same preferences? Explain. 5. Suppose X, Y, Z are metric spaces and f : X Y and g : Y Z are continuous functions. Prove that g f : X Z is continuous. [Hint: You can use any characterization of continuity.] 2. Let V be a vector space. (a) Define a subspace S of V. (b) Let S and T be subspaces of V. Prove that S + T is a subspace of V. 3. Suppose the function f : R 2 R 2 is given by (y 1, y 2 ) = (x 1 cos θ x 2 sin θ, x 1 sin θ + x 2 cos θ) for 0 θ 2π. (a) Is f a bijection? Explain. (b) If θ = π/2, what is f 1 (0, 1)? (c) Find the fixed point(s) of f for θ = π/2.!
Test 4 November 11, 2005 Read Me: Please write your answers on the answer book provided. Use the right-side pages for formal answers and the left-side pages for your rough work. Do not forget to put your name on the front page. 1. Suppose that a linear operation f on R 3 is represented by the matrix 0 0 0 A = 0 1 0. 0 0 1 with respect to the standard basis. (a) Identify the kernel of f. (b) What is the nullity of f? (c) What is the rank of f? (d) Do your results above conform with the Dimension Theorem? Explain. 2. Suppose that f and g are two invertible linear operators on a vector space V. Prove that for all x V. (g f) 1 (x) = (f 1 g 1 )(x) (a) Find the eigenvalues and the normalized eigenvectors of A. (b) Determine the definiteness of A. (c) Find the eigenvalues and the normalized eigenvectors of A 1. 4. (a) State the Spectral Theorem (diagonal decomposition). (b) Prove that the product of the eigenvalues of a symmetric matrix A is equal to its determinant, that is, n λ i = A. i=1 5. Let x 1 and x 2 be two real numbers. Show that (x 1 x 2 ) 1/2 1 2 (x 1 + x 2 ), that is, the geometric mean is less than or equal to the arithmetic mean. [Hint: Let y 1 = log x 1, y 2 = log x 2 and use the fact that e x is convex.] 3. Suppose that A = ( 1 2 2 1 ).
Test 5 November 25, 2005 Read Me: Please write your answers on the answer book provided. Use the right-side pages for formal answers and the left-side pages for your rough work. Do not forget to put your name on the front page. 1. Let f(x 1, x 2 ) = x 1/3 1 x 2/3 2. (a) Find the gradient of f. (b) Find the directional derivative in the direction of e 2. 2. Prove the Cauchy-Schwarz Inequality: Let x, y R n. Then (x T y) 2 (x T x)(y T y) M/P = L(Y, r). In this model, Y (output) and r (interest rate) are endogenous and P (price), G (government expenditure), t (tax rate), and M (money demand) are exogenous. C, I, and L are functions for consumption, investment, and money demand respectively. (a) Find the output multiplier for income tax, Y/ t. (b) Investigate the effect of inflation on the interest rate. (c) What assumptions do you have to make in order for the model to work? Hint: Assume x and y 0, and therefore (x T x) > 0 and (y T y) > 0. Now for every real number α, define z = x + αy so that z R n. Find the minimum value of z T z. 3. Prove that the gradient of a differentiable functional f points in the direction of greatest increase. Hint: Use the Cauchy-Schwarz Inequality. 4. Suppose f(x) = (x 1 + x 2 ) 2. (a) Does f 1 exist in the neighbourhood of the origin? (b) Find the second order Taylor formula for f about the origin. 5. Consider the following Keynesian model in macroeconomics: Y = C[(1 t)y ] + I(r, Y ) + G,
Final Examination (RB1044) December 5, 2005 Time Allowed: 2 hour Instruction: Please write your answers on the answer book provided. Use the right-side pages for formal answers and the left-side pages for your rough work. Also, start each question on a new page. Read the questions carefully and provide answers to what you are asked only. Do not spend time on what you are not asked to do. Remember to put your name on the front page. 1. Let f : R n + R + be a homogeneous function of degree k. (a) If k = 1, show that f(x) = f(x) T x. (b) If n = 1, show that f is a multiple of a power function, that is, f(x) = Ax k for some A R. 2. Consider the function (a) Find the ordinary demand function for the two products. (b) Define the indirect utility function V (p 1, p 2, M) = u(x 1, x 2) to be the value function of the optimal choice. Show that V (p 1, p 2, M) M = λ, where λ is the Lagrange multiplier. 5. Solve the following maximization problem: max x,y 2x + log(3 + y) subject to x + y 1, x 0, y 0. f(x 1, x 2 ) = 3x 1 x 2 x 3 1 x 3 2. (a) Find all the stationary points of f. (b) Determine whether each point is a local maximum, a local minimum, or a saddle point. 3. Suppose that y = Xβ + ɛ where y R n, X is a n k matrix, and ɛ R n is the vector of random errors. (a) Derive an expression for ɛ T ɛ. (b) Assuming X have rank k, derive the least square estimator for β. (c) Check the second order condition. 4. Suppose a consumer maximizes her utility by buying two products with amounts x 1 and x 2 according to the utility function u(x 1, x 2 ) = x α 1 x 1 α 2. Her budget constraint is p 1 x 1 + p 2 x 2 = M where p 1, p 2 are the prices and M is her weekly expenditure on the two products.