Problem Sheet 11 MT1100 From Euclid to Mandelbrot Not marked: Use solutions on Moodle to check your answers 1. (i) Show that [0, 1]| = [1, a]|, where a is any constant a > 1, by writing down a bijective function f : [0, 1] > [1, a]. (ii) Let A = [0, 1) U [1, 2) and B = [0, 1). Show that | A| = | B| by writing down a bijective function f : A -> B. 2. Find four pairs of infinite sets A and B, each pair satisfying one of the following properties: (i) A C B and A + B and | A| = | B|; (ii) AUB = N, with A + B, and | A| = | B|; (iii) AnB = 0, ACN, B CN and | A| = | B| = No; (iv) AUB = R and | A| < |B| = c, where c = |R|. 3. Let A and B be sets such that A C B. By writing down an injection from A to B, show that |A| ? | B|. 4. Show that |(0, 1)| = [0, 1]|. [Hint: Show that |(0, 1)| ? |[0, 1]| and |[0, 1]| ? |(0, 1)|. Then use the Cantor-Bernstein-Schroeder Theorem.] 5. Let T be the set of all infinite ternary sequences a1, a2, a3 . .. where ai E {0, 1, 2}. Use a diagonal argument to show that T is uncountable. 1