Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
5. Let x and y be irrational numbers such that x-y is also irrational. Define sets A and B by A = {x +r:reQ} and B = {y+r:reQ). Prove that the sets A and B have no elements in common. be the set of all numbers of the form a t b/2 where a and b are arbitrary

5. Let x and y be irrational numbers such that x-y is also irrational. Define sets A and B by A = {x +r:reQ} and B = {y+r:reQ). Prove that the sets A and B have no elements in common. be the set of all numbers of the form a t b/2 where a and b are arbitrary

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
**5. Let \( x \) and \( y \) be irrational numbers such that \( x - y \) is also irrational. Define sets \( A \) and \( B \) by \( A = \{ x + r : r \in \mathbb{Q} \} \) and \( B = \{ y + r : r \in \mathbb{Q} \} \). Prove that the sets \( A \) and \( B \) have no elements in common.** **Explanation:** The problem involves showing that two sets constructed by adding rational numbers to given irrational numbers are disjoint, under the condition that the difference between the irrational numbers is also irrational. Here’s the detailed reasoning: 1. **Definitions of Sets:** - Set \( A = \{ x + r : r \in \mathbb{Q} \} \): This represents all numbers formed by adding any rational number \( r \) to the irrational number \( x \). - Set \( B = \{ y + r : r \in \mathbb{Q} \} \): This represents all numbers formed by adding any rational number \( r \) to the irrational number \( y \). 2. **Condition:** - \( x - y \) is irrational. 3. **Proof Strategy:** - Assume for contradiction that there exists some element common to both sets \( A \) and \( B \). - Then there exist rational numbers \( r_1 \) and \( r_2 \) such that \( x + r_1 = y + r_2 \). - Rearranging gives \( x - y = r_2 - r_1 \), which implies that \( x - y \) is rational, a contradiction. Therefore, the sets \( A \) and \( B \) have no elements in common.
Transcribed Image Text:**5. Let \( x \) and \( y \) be irrational numbers such that \( x - y \) is also irrational. Define sets \( A \) and \( B \) by \( A = \{ x + r : r \in \mathbb{Q} \} \) and \( B = \{ y + r : r \in \mathbb{Q} \} \). Prove that the sets \( A \) and \( B \) have no elements in common.** **Explanation:** The problem involves showing that two sets constructed by adding rational numbers to given irrational numbers are disjoint, under the condition that the difference between the irrational numbers is also irrational. Here’s the detailed reasoning: 1. **Definitions of Sets:** - Set \( A = \{ x + r : r \in \mathbb{Q} \} \): This represents all numbers formed by adding any rational number \( r \) to the irrational number \( x \). - Set \( B = \{ y + r : r \in \mathbb{Q} \} \): This represents all numbers formed by adding any rational number \( r \) to the irrational number \( y \). 2. **Condition:** - \( x - y \) is irrational. 3. **Proof Strategy:** - Assume for contradiction that there exists some element common to both sets \( A \) and \( B \). - Then there exist rational numbers \( r_1 \) and \( r_2 \) such that \( x + r_1 = y + r_2 \). - Rearranging gives \( x - y = r_2 - r_1 \), which implies that \( x - y \) is rational, a contradiction. Therefore, the sets \( A \) and \( B \) have no elements in common.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS