Use class Problem 4. Prove that F = (FnEjv (FnE") and Eu F) = EUE" n...
Please note n's are superscripted. (a) Use mathematical induction to prove that 2n+1 + 3n+1 ≤ 2 · 4n for all integers n ≥ 3. (b) Let f(n) = 2n+1 + 3n+1 and g(n) = 4n. Using the inequality from part (a) prove that f(n) = O(g(n)). You need to give a rigorous proof derived directly from the definition of O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how f(n) =...
4. (5 points) Use the substitution method to prove the guess that is indeed correct when T(n) is defined by the following recurrence relations: T(n) = 3T(n/3) +5; T(1) = 5. At the end of your proof state the value of constant c that is needed to make the proof work. Statement of what you have to prove: Base Case proof: Inductive Hypotheses: Inductive Step: Value of c: 5. (6 points) Find a counterexample to the following claim: f(n)=O(s(n)) and...
state any definitions or theorems used
Question 2. In this problem we'll prove that if a<b<c and f is integrable on [a, cl ther it's also integrable on [a,b] and [b, c'. Our approach will be to show that for all ε > 0 there are partitions Q1 and Q2 of [a, b) and [b, c] respectively with Thus, let ε > 0 be given. By our fundamental lemma there exists a partition P of [a, c) such that U...
+20 Problem 7. Let f :D + R, xo be an accumulation point of D and assume lim f(x) = L. Use the e-8 definition of the limit (not theorems or results from class or the text) to prove the following: (a) The function f is “bounded near xo”: there is an M ER and a 8 >0 such that for x E D, 0 < l< – xo<8 = \f(x) < M. Hint: compare with the proof that a...
(4) (1 point) PFnGn He). Problem 2 (3 points) Use proof by induction to prove the Boole's inequality (for any positive integer n): TI 7l i -1
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Problem 11. Prove via induction that every integer n 2 can be expressed as a product of prime mumbers. You may use without proof that if n 2 2 is no such that n ab. t prime, then there exists integers a, b2 2
Problem 11. Prove via induction that every integer n 2 can be expressed as a product of prime mumbers. You may use without proof that if n 2 2 is no such that n ab. t...
Problem 3 (3 points) Use proof by induction to prove the Bonferroni's inequality (for any positive integer n): Si<jSni.jez
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
Formal Definitions of Big-Oh, Big-Theta and Big-Omega:
1. Use the formal definition of Big-Oh to prove that if f(n) is a decreasing function, then f(n) = 0(1). A decreasing function is one in which f(x1) f(r2) if and only if xi 5 r2. You may assume that f(n) is positive evervwhere Hint: drawing a picture might make the proof for this problem more obvious 2. Use the formal definition of Big-Oh to prove that if f(n) = 0(g(n)) and g(n)...