MTH560 FALL 2017, ASSIGNMENT 2 Last Revised1 Oct 28, 2017 Due at 12:10 pm in your class on T.B.A, November ??. Very late assignments will not be accepted. Slightly late assignments will have a mark reduction made. If you are unable to complete the assignment, just hand in to Prof. Grandison what you have at the deadline. 1. f : D C is a function with domain D and codomain C. (a) Prove that if A D then A f 1 (f (A)). (b) Show that f is one to one if and only if for every choice of subsets A1 and A2 from D we have f (A1 A2 ) = f (A1 ) f (A2 ). (c) When can we say f (f 1 (B)) = B. 2. Prove that if 0 < |h| < 1 then 1 + h < eh < 1 . 1h sin h then 1 + h2 < <1 2 h 4. Prove that if 0 < |h| < then 0 < 1 cos h < h sin h < h2 . 2 3. Prove that if 0 < |h| < 5. Prove2 that for all real numbers a, b and c the following (triangle inequality) applies: |a c| |a b| + |b c|. 6. Recall that a set F in a metric space (X, d) is closed means that F contains all its limit points. Prove that a set is closed if and only if it is the compliment of an open set. Please be sure your assignments are stapled, your name, the assignment number and date are clearly printed on them, otherwise you may incur a mark penalty. 1 This document was generated from source on Sunday 29th October, 2017 at 21:59. Use ( the following definition for the absolute value of a real number. x if 0 x |x| = x if x < 0 2