assn3t - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER...

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Unformatted text preview: CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING COMP 232/2 Mathematics for Computer Science FALL 2010 Assignment 3 1. Let a, b, c R with b 6 = ac and let the function f : R R be given by f ( x ) = 8 < : a if x = c, ax- b x- c if x 6 = c. i) Show that f is one-to-one. ii) Show that f is onto. iii) Find f- 1 ( y ). 2. For sets A , B , and S , with S B , and a function f : A B , we define f- 1 ( S ) = a A : f ( a ) S . For each of the following find f- 1 ( S ). A B f S i) R R x 7 x- b x c { y : 0 < y < 1 } ii) R R x 7 x 3- 7 x + 16 { y : 10 y 22 } iii) R R R t 7 (cos t, sin t ) { ( x, y ) : x < 0, y > } 3. Let x be a real number. i) Prove that b- x c =-d x e and d- x e =-b x c . ii) Give a proof by cases that 4 x = x + x + 1 4 + x + 1 2 + x + 3 4 . 4. Let a and b be relatively prime positive integers....
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