Semester Problem Set 2 Discrete Math Solutions
ATILIM UNIVERSITY DEPARTMENT OF MATHEMATICS Math 211 - Discrete Mathematics with Applications 2010-2011 Fall Semester Problem Set I Prepared by Mehmet TURAN
O−, Ω−, Θ− Notations
1. Let f and g be real valued functions defined on the same set of nonnegative real numbers. (a) Prove that if g(x) is O(f (x)), then f (x) is Ω(g(x)). (b) Prove that if f (x) is O(g(x)) and c is any nonzero ral number, then cf (x) is O(cg(x)). (c) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x) + g(x) is O(G(x)), where, for each x in the domain, G(x) = max(|h(x)|, |k(x)|). (d) Prove that f (x) is Θ(f (x)). (e) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x)g(x) is O(h(x)k(x)). 2. (a) Show that for any real number x, if x > 1 then |x3
O−, Ω−, Θ− Notations
1. Let f and g be real valued functions defined on the same set of nonnegative real numbers. (a) Prove that if g(x) is O(f (x)), then f (x) is Ω(g(x)). (b) Prove that if f (x) is O(g(x)) and c is any nonzero ral number, then cf (x) is O(cg(x)). (c) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x) + g(x) is O(G(x)), where, for each x in the domain, G(x) = max(|h(x)|, |k(x)|). (d) Prove that f (x) is Θ(f (x)). (e) Prove that if f (x) is O(h(x)) and g(x) is O(k(x)), then f (x)g(x) is O(h(x)k(x)). 2. (a) Show that for any real number x, if x > 1 then |x3