(a) Give a proof by contradiction of: for all x € R, if x² is irrational, then x is irratio (b) Give a proof by contraposition of: for all integers n, if 3n - 5 is even, then n is o (c) Give a proof by cases of: x ≤ |x| for all real numbers x. (d) Give a direct proof of: if x and y are odd integers, then x + y is even.” Lo 1 I 1 Lal 11

(a) Give a proof by contradiction of: for all x € R, if x² is irrational, then x is irratio (b) Give a proof by contraposition of: for all integers n, if 3n - 5 is even, then n is o (c) Give a proof by cases of: x ≤ |x| for all real numbers x. (d) Give a direct proof of: if x and y are odd integers, then x + y is even.” Lo 1 I 1 Lal 11

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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Concept explainers

Problems on numbers

Number theory is one of the most appealing topics in mathematics. Given one or more numeric values, a numeric algorithm performs some computation. The use of number theory in computing is important for applications like in the case of cryptographic algorithms. Number theory is also used in various applications such as calculating hash functions or in information security. Euclid’s algorithm is one of the oldest number-theoretic algorithms which is used to calculate the greatest common divisor of two numbers.

Question

PS: Please complete all of the subparts as they are all the same question and not in Handwritten paper.

5. (a) Give a proof by contradiction of: for all x € R, if x² is irrational, then x is irrational.
(b) Give a proof by contraposition of: for all integers n, if 3n – 5 is even, then n is odd.
(c) Give a proof by cases of: x ≤ |x| for all real numbers x.
(d) Give a direct proof of: if x and y are odd integers, then x + y is even."
(e) Prove or disprove: for all real numbers x and y, [x − y] = [x] − [y]

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