1. Prove the following propositions: (a) For all real numbers x and y, if x is rational and y is irrational, then x+y is irrational. (b) For all nonzero real numbers x and y, if x is rational and y is irrational, then x/y is irrational. 2. If n is a positive integer, then nª-n is divisible by 4. [Proof of Exhaustion] 3. There are no positive integers a and b such that 18a + 6b = 1. (a) Write this statement as: if P then 18a + 6b/ 1. (b) What are the first two sentences of your contradiction proof of this claim? (c) Find a contradiction. (d) Conclude the proof.
1. Prove the following propositions: (a) For all real numbers x and y, if x is rational and y is irrational, then x+y is irrational. (b) For all nonzero real numbers x and y, if x is rational and y is irrational, then x/y is irrational. 2. If n is a positive integer, then nª-n is divisible by 4. [Proof of Exhaustion] 3. There are no positive integers a and b such that 18a + 6b = 1. (a) Write this statement as: if P then 18a + 6b/ 1. (b) What are the first two sentences of your contradiction proof of this claim? (c) Find a contradiction. (d) Conclude the proof.
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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![1. Prove the following propositions:
(a) For all real numbers x and y, if x is rational and y is irrational,
then x+y is irrational.
(b) For all nonzero real numbers x and y, if x is rational and y is
irrational, then x/y is irrational.
2. If n is a positive integer, then n4 - n is divisible by 4. [Proof of
Exhaustion]
3. There are no positive integers a and b such that 18a + 6b = 1.
(a) Write this statement as: if P then 18a + 6b 1.
(b) What are the first two sentences of your contradiction proof
of this claim?
(c) Find a contradiction.
(d) Conclude the proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0fce20d-b8b1-4b07-858d-626ada44a0c6%2F84b37da0-b3fb-459d-abdc-3ccaea2e47a1%2Fs8mul6m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Prove the following propositions:
(a) For all real numbers x and y, if x is rational and y is irrational,
then x+y is irrational.
(b) For all nonzero real numbers x and y, if x is rational and y is
irrational, then x/y is irrational.
2. If n is a positive integer, then n4 - n is divisible by 4. [Proof of
Exhaustion]
3. There are no positive integers a and b such that 18a + 6b = 1.
(a) Write this statement as: if P then 18a + 6b 1.
(b) What are the first two sentences of your contradiction proof
of this claim?
(c) Find a contradiction.
(d) Conclude the proof.
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