Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Is the set X of all functions f: Z+→ {0, 1} countable? Prove or disprove.

Is the set X of all functions f: Z+→ {0, 1} countable? Prove or disprove.

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
**Question:** Is the set \( X \) of all functions \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) countable? Prove or disprove. --- To determine if the set \( X \) of all functions from the set of positive integers \( \mathbb{Z}^+ \) to the set \(\{0,1\} \) is countable, we first need to understand some key concepts: 1. **Countable Sets:** - A set is **countably infinite** if its elements can be placed in one-to-one correspondence with the positive integers \( \mathbb{Z}^+ \) (i.e., it can be listed as \( \{a_1, a_2, a_3, \ldots\} \)). - A set is **uncountable** if it is not countable (i.e., it cannot be listed in this manner). 2. **Functions:** - A function \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) assigns to each positive integer either 0 or 1. - Each function can be represented as an infinite sequence of 0s and 1s. For example, one such function might look like \( f(1)=0, f(2)=1, f(3)=0, \ldots \), which corresponds to the sequence \( (0, 1, 0, \ldots) \). 3. **Cardinality:** - The set of all infinite sequences of 0s and 1s has the same cardinality as the power set of the natural numbers \( \mathcal{P}(\mathbb{N}) \). - The cardinality of \( \mathcal{P}(\mathbb{N}) \) is \( 2^{\aleph_0} \) (read as \( 2 \) raised to the power of \( \aleph_0 \), where \( \aleph_0 \) denotes the cardinality of the set of positive integers \( \mathbb{Z}^+ \), representing the smallest infinity, also known as countable infinity). **Proof:** 1. Assume \( X \) is countable. Then, there would exist a bijection between the set of all functions \(
Transcribed Image Text:**Question:** Is the set \( X \) of all functions \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) countable? Prove or disprove. --- To determine if the set \( X \) of all functions from the set of positive integers \( \mathbb{Z}^+ \) to the set \(\{0,1\} \) is countable, we first need to understand some key concepts: 1. **Countable Sets:** - A set is **countably infinite** if its elements can be placed in one-to-one correspondence with the positive integers \( \mathbb{Z}^+ \) (i.e., it can be listed as \( \{a_1, a_2, a_3, \ldots\} \)). - A set is **uncountable** if it is not countable (i.e., it cannot be listed in this manner). 2. **Functions:** - A function \( f : \mathbb{Z}^+ \rightarrow \{0,1\} \) assigns to each positive integer either 0 or 1. - Each function can be represented as an infinite sequence of 0s and 1s. For example, one such function might look like \( f(1)=0, f(2)=1, f(3)=0, \ldots \), which corresponds to the sequence \( (0, 1, 0, \ldots) \). 3. **Cardinality:** - The set of all infinite sequences of 0s and 1s has the same cardinality as the power set of the natural numbers \( \mathcal{P}(\mathbb{N}) \). - The cardinality of \( \mathcal{P}(\mathbb{N}) \) is \( 2^{\aleph_0} \) (read as \( 2 \) raised to the power of \( \aleph_0 \), where \( \aleph_0 \) denotes the cardinality of the set of positive integers \( \mathbb{Z}^+ \), representing the smallest infinity, also known as countable infinity). **Proof:** 1. Assume \( X \) is countable. Then, there would exist a bijection between the set of all functions \(
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
Similar questions
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