1 Prove the following using the definitions of the notations, or disprove with a specific counterexample:...
7. Prove the following assertions by using the definitions of the notations in- volved, or disprove them by giving a specific counterexample. a. If t(n) e (g(n)), then g(n) E S2(t(n I. Θ(gg(n))-e(g(n)), where α > 0. c. Θ(g(n))-: 0(g(n))n Ω (g(n)). d. For any two nonnegative functions t (n) and g(n) defined on the set of nonnegative integers, either t (n) e 0(g(n)), or t (n) e Ω(g(n)), or both 7. Prove the following assertions by using the definitions...
Let f1 and f2 be asymptotically positive non-decreasing functions. Prove or disprove each of the following conjectures. To disprove, give a counterexample. a.If f1(n) = Theta(g(n)) and f2(n) = Theta(g(n)) then f1(n) + f2(n) = Theta(g(n)) b.If f1(n) = O(g(n)) and f2(n) = O(g(n))then f1(n) = O(f2(n))
Suppose f(n) = O(s(n)), and g(n) = O(r(n)). All four functions are positive-valued and monotonically increasing. Prove (using the formal definitions of asymptotic notations) or disprove (by counterexample) each of the following claims: (a) f(n) − g(n) = O(s(n) − r(n)) (b) if s(n) = O(g(n)), then f(n) = O(r(n)) (c) if r(n) = O(s(n)), then g(n) = O(f(n)) (d) if s(n) + g(n) = O(f(n)), then f(n) = Θ(s(n))
Prove or find a counterexample for the following. Assume that f (n) and g (n) are monotonically increasing functions that are always larger than 1. f (n) = o (g (n)) rightarrow log (f (n)) = o (log (g (n))) f (n) = O (g (n)) rightarrow log (f (n)) = O (log (g (n))) f (n) = o (g (n)) rightarrow 2^f (n) = o (2^g (n)) f (n) = O (g (n)) rightarrow 2^f (n) = O (2^g...
Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n)), and g(n) = O(n2), then f(n) + g(n) = O(n5). Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n), and g(n) = O(n2), then fin)/g(n) = O(n).
Need help with 1,2,3 thank you. 1. Order of growth (20 points) Order the following functions according to their order of growth from the lowest to the highest. If you think that two functions are of the same order (Le f(n) E Θ(g(n))), put then in the same group. log(n!), n., log log n, logn, n log(n), n2 V, (1)!, 2", n!, 3", 21 2. Asymptotic Notation (20 points) For each pair of functions in the table below, deternme whether...
Help please! Using matlab Prove or give a counterexample: if f: X rightarrow Y and g: Y rightarrow X are functions such that g o f = I_X and f o g = I_Y, then f and g are both one-to-one and onto and g = f^-1.
For each of the following statements, either prove it is true, or provide a counterexample to show that it is false. (a) If (sn) is a sequence such that lim sn = 0, then lim inf|sn= 0. (b) If f : [0, 1] + R is a function with f(0) < 0 and f(1) > 0, then there exists CE (0,1) such that f(c) = 0. (c) If I is an interval, f:I + R is continuous on I, and...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Prove or disprove the following statements, using the relationship among typical growth-rate functions seen in class. a)n^15log n + n^9 is O(n^9 log n) b) 15^7n^5 + 5n^4 + 8000000n^2 + n is Θ(n^3) c) n^n is Ω (n!) d) 0.01n^9 + 800000n^7 is O(n^9) e) n^14 + 0.0000001n^5 is Ω(n^13) f) n! is O(3n)