Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
real numbers such that a < b.

real numbers such that a < b.

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
**Real Analysis Problem: Uncountable and Countably Infinite Sets** **Problem Statement:** Suppose that \( a \) and \( b \) are distinct real numbers such that \( a < b \). a) Prove that the set \(\{x \in \mathbb{R} : a < x < b\}\) is uncountable. b) Prove that the set \(\{x \in \mathbb{Q} : a < x < b\}\) is countably infinite. **Key Concepts:** - **Real Numbers (\(\mathbb{R}\))**: The set of all points on the number line, including both rational and irrational numbers. - **Rational Numbers (\(\mathbb{Q}\))**: The set of numbers that can be expressed as a fraction of two integers. - **Uncountable Set**: A set that contains more elements than the set of natural numbers; cannot be put into a one-to-one correspondence with the natural numbers. - **Countably Infinite Set**: A set that can be put into a one-to-one correspondence with the natural numbers. **Analysis:** - For part (a), we demonstrate the uncountability of \(\{x \in \mathbb{R} : a < x < b\}\) by leveraging the properties of real numbers and perhaps using Cantor's diagonal argument or the fact that any interval in \(\mathbb{R}\) has the same cardinality as \(\mathbb{R}\) itself. - For part (b), we show that \(\{x \in \mathbb{Q} : a < x < b\}\) is countably infinite by finding a bijective relation with \(\mathbb{N}\), the set of natural numbers, using the density of rationals in the reals.
Transcribed Image Text:**Real Analysis Problem: Uncountable and Countably Infinite Sets** **Problem Statement:** Suppose that \( a \) and \( b \) are distinct real numbers such that \( a < b \). a) Prove that the set \(\{x \in \mathbb{R} : a < x < b\}\) is uncountable. b) Prove that the set \(\{x \in \mathbb{Q} : a < x < b\}\) is countably infinite. **Key Concepts:** - **Real Numbers (\(\mathbb{R}\))**: The set of all points on the number line, including both rational and irrational numbers. - **Rational Numbers (\(\mathbb{Q}\))**: The set of numbers that can be expressed as a fraction of two integers. - **Uncountable Set**: A set that contains more elements than the set of natural numbers; cannot be put into a one-to-one correspondence with the natural numbers. - **Countably Infinite Set**: A set that can be put into a one-to-one correspondence with the natural numbers. **Analysis:** - For part (a), we demonstrate the uncountability of \(\{x \in \mathbb{R} : a < x < b\}\) by leveraging the properties of real numbers and perhaps using Cantor's diagonal argument or the fact that any interval in \(\mathbb{R}\) has the same cardinality as \(\mathbb{R}\) itself. - For part (b), we show that \(\{x \in \mathbb{Q} : a < x < b\}\) is countably infinite by finding a bijective relation with \(\mathbb{N}\), the set of natural numbers, using the density of rationals in the reals.
Expert Solution
Step 1

Advanced Math homework question answer, step 1, image 1

steps

Step by step

Solved in 4 steps with 4 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