Set Theory By Naylor and Sell Homework Problems

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Homework Help Overview

The discussion revolves around understanding problems related to set theory, specifically focusing on the cardinality of sets of natural numbers and even natural numbers as presented in a textbook on Linear Operator Theory. Participants are exploring the implications of various mappings between these sets.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are examining different mappings between the set of even natural numbers and the set of natural numbers, questioning whether these mappings are one-to-one and onto. There is a discussion about the implications of these mappings on the cardinality of the sets involved.

Discussion Status

Some participants have provided mappings and are discussing their properties, with one participant suggesting that the cardinality of the two sets is equal based on their mapping. However, there is no explicit consensus, as others are still exploring the implications of the mappings.

Contextual Notes

There is a mention of a specific textbook and the need for clarity on the exercise problems presented within it. Some participants express a lack of access to the textbook, which may limit the depth of discussion.

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Homework Statement



I am trying understand the approach to the problems given in the following textbook. Linear Operator Theory in Engineering and Science (Applied Mathematical Sciences) (Volume 0) [Paperback]
Arch W. Naylor (Author), George R. Sell (Author). Pages 12-29. There are exercise problems for which I need approach. Please let me know if anyone has a good understanding of them. Appreciate your response.

Homework Equations





The Attempt at a Solution

 
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Speaking for myself, I don't have this book. If you have questions about one of the problems, post it here, along with your attempt at a solution.
 
Is Card (N even)< Card (N)? Where N even is set of all even Natural numbers, N is set of all Natural numbers.

Hint: use the mapping from
N even to N is given by n-->n
a. Show examples of this mapping from
N even to N.

b. Is the mapping above onto? One-to-one?

My try at this question:

Since the mapping is n-->n it is the range of this mapping function is a proper subset of N hence the card (N even) <card (N) because there is one to one mapping and the N even maps into N.

Examples above mapping. 2->2, 4->4, 8->8, 268->268... Hence card(N even)< Card
 
How about this mapping:
2 --> 1
4 --> 2
6 --> 3
8 --> 4
.
.
.
2k --> k
This mapping is 1-to-1 and onto. What does that say about the cardinality of the two sets?
 
Mark44 said:
How about this mapping:
2 --> 1
4 --> 2
6 --> 3
8 --> 4
.
.
.
2k --> k
This mapping is 1-to-1 and onto. What does that say about the cardinality of the two sets?

They are equal!
 

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