c) What is the inductive hypothesis of a proof that P(n) is true for all positive integers n? d) What do you need to prove in the inductive step of a proof that P(n) is true for all positive integers n? e) Complete the inductive step of a proof that P(n) is true for all positive integers n, identifying where you use the inductive hypothesis.

c) What is the inductive hypothesis of a proof that P(n) is true for all positive integers n? d) What do you need to prove in the inductive step of a proof that P(n) is true for all positive integers n? e) Complete the inductive step of a proof that P(n) is true for all positive integers n, identifying where you use the inductive hypothesis.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Let P(n) be the statement that a postage of n cents can be formed using just 4-cent and 5-cent stamps." Use strong mathematical induction to prove that P(n) is true for n ≥ 12. Answer the following questions to show a complete proof (Please remember that you must prove the statement using strong mathematical induction) Show that the statements P(12), P(13), P(14), and P(15) are true, completing the basis step of the proof. What is the inductive hypothesis of the proof? What do you need to prove in the inductive step? Complete the inductive step for k ≥ 15.
PS: Please complete all of the subparts as they are all the same question and not in Handwritten paper.
= For integer n ≥ 1 let P(n) be the predicate that 9" - 5n CEZ 4c for some For the induction hypothesis, consider k ≥ 1, and suppose that P(k) is true. For the inductive step, we want to show that P(k+1) is true. True or false: The following proof correctly proves P(k+ 1) true, where every step other than the one labelled IH follows by algebra. (I'm asking: is this a valid algebraic proof? Is the algebra correct? Did I use the IH correctly? Did I get the correct final result?) 9k+15k+19.9k - 5.5k True False = 9(4c – 5k) – 5 ⋅ 5% for CE Z by the IH = 4.9c-9.5k - 5.5k = 4.9c-4.5k = 4d for d = Z
This is NON GRADED QUESTION Proof of Correctness In this question you will use strong induction to prove that your new algorithm works correctly. In other words, you will prove that "nÎℕ "xÎℝ-{0} FP(x,n) = xn 1. Predicate Function Your conjecture has already been stated in symbolic form: It is a statement of the form "nÎℕ, P(n) What is the predicate function P(n)? a) Proof: Base cases b) Proof: Inductive step setup This is the beginning of the inductive step where you are stating the assumptions in the inductive step and what you will be proving in that step. As you do so, identify the inductive hypothesis. c) Proof: Inductive step
1. Prove by induction that Vi > 1, (L*)' = L*. • Basis: ? • Inductive Hypothesis: ? • Complete the rest of the proof: ?
Q2 – Proof of Correctness In this question, you will use strong induction to prove that your new algorithm works correctly. In other words, you will prove that nN xR-{0} FP(x,n) = xn a) Predicate Function Your conjecture has already been stated in symbolic form: It is a statement of the form nN, P(n) What is the predicate function P(n)? b) Proof: Base cases c) Proof: Inductive step setup This is the beginning of the inductive step where you are stating the assumptions in the inductive step and what you will be proving in that step. As you do so, identify the inductive hypothesis. d) proof: inductive steps
This is Discreet math please answer questions with proper formatting for proofs
Write a formal proof of the following statement. Be sure to copy this statement on your paper before beginning the proof. For all integers n, n is even if and only if 3n is even. Note: An "if-and-only-if" statement requires that you prove both implications: "if n is even then 3n is even" and "if 3n is even then n is even."
Topic: Proofs (even (2k), odd(2k+1), r = a / b. etc.) Let a and b be rational numbers such that a > b. Then there exists a rational number c such that a > c > b.
A prime number is a natural number greater than 1 which is not a product of two smaller natural numbers. Prove or disprove: For every integer q, if q > 7, then q can be written as q = (a + b * c) such that the following properties hold a and b are prime numbers, c is an integer greater than 1.
3. Let P(n) be the statement that 1² +2²+...+ n² = n(n+ 1)(2n + 1)/6 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of a proof that P(n) is true for all positive integers n. c) What is the inductive hypothesis of a proof that P(n) is true for all positive integers n? d) What do you need to prove in the inductive step of a proof that P(n) is true for all positive integers n? e) Complete the inductive step of a proof that P(n) is true for all positive integers n, identifying where you use the inductive hypothesis.
2- Prove thatthe following statements are true (7) or false (F). (let log n = log₂n). You must define first what you are trying to prove using the limit definition. a) n³/logn O(n), for any integer constant 2 ≤k< 4. b) n+nlogne (n²), for any positive integer constant k.