M AT H 1 2 0 Differential and Integral Calculus Winter 2016 - Problem Set #17 Due Thursday, February 25th. Write your name and student number very clearly in the upper right hand corner of your front sheet and staple your sheets if necessary. You are encouraged to collaborate with others when working on your homework assignments, but you must write up solutions independently, on your own. Do not copy the work of others. In accordance with academic integrity regulations, you must acknowledge in writing the assistance of any students, professors, books, calculators, or software. Be sure that your nal write-up is clean and clear and effectively communicates your reasoning to the grader. 1) Consider the function from the closed interval [0, 1] to the open interval (0, 1). 1 if x = 0, 2 1 f (x) = n+2 if x = 2n for some n Z, 2 x otherwise. (a) Show that f is injective. (b) Show that f is surjective. 2) The purpose of this problem is to nd a bijection from the closed disk D = {( x, y) R2 | x2 + y2 1} to the circle C = {( x, y) R2 | x2 + y2 = 1}. (a) Find a bijection f 1 from the closed disk D to the closed square [0, 1] [0, 1] = [0, 1]2 . (b) Find a bijection f 2 from [0, 1]2 to the closed interval [0, 1]. (c) Find a bijection f 3 from the closed interval [0, 1] to the half-open interval [0, 1). [Hint: nd inspiration in problem 1.] (d) Find a bijection f 4 from [0, 1) to the circle C . (e) Use (a) (d) to nd a bijection f from the disk D to the circle C . Remark: this procedure is certainly not the only way to nd a bijection from D to C . turn the page 3) A real number is called algebraic if it is the root of a polynomial with rational coefcients. 3 For example, the number 7 is algebraic because it is the root of x 7 = 0, the number 19 is algebraic because it is the root of 19x + 3 = 0, the number 2 is algebraic because it is the root of x2 2 = 0, the number 2 + 2 is algebraic because it is the root of x4 4x2 + 2 = 0, the number cos( ) is algebraic 7 2 because it is the root of 3 4x 4x + 1 = 0. 8x + Numbers such as , e, e 2 , e , 2 2 , sin(1), cos( 217 3 ), ln(8), ln( 6 7), ... are all not algebraic, but this is far from obvious (it is a consequence of a difcult theorem due to Lindemann and Weierstrass)! Real numbers that are not algebraic are called transcendental. (a) Show that if is algebraic, then it is the root of a polynomial with integer coefcients. (b) Show that the set of polynomials with integer coefcients and xed degree, say d, is countable. (c) Use (a) (b) to prove that the set of algebraic numbers is countable. [Hint: it may be useful to also use an argument similar to the one we used in class to show that Q+ is countable.] (d) Is the set of transcendental numbers countable? Is it uncountable? Explain your