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1 - Theory of sets

Published online by Cambridge University Press:  21 March 2010

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Summary

Sets

We do not want to become involved in the logical foundations of mathematics. In order to avoid these we will adopt a rather naive attitude to set theory. This will not lead us into difficulties because in any given situation we will be considering sets which are all contained in (are subsets of) a fixed set or space or suitable collections of such sets. The logical difficulties which can arise in set theory only appear when one considers sets which are ‘too big’—like the set of all sets, for instance. We assume the basic algebraic properties of the positive integers, the real numbers, and Euclidean spaces and make no attempt to obtain these from more primitive set theoretic notions. However, we will give an outline development (in Chapter 2) of the topological properties of these sets.

In a space X a set E is well defined if there is a rule which determines, for each element (or point) x in X, whether or not it is in E. We write xE (read ‘x belongs to E’) whenever x is an element of E, and the negation of this statement is written xE. Given two sets E, F we say that E is contained in F, or E is a subset of F, or F contains E and write EF if every element x in E also belongs to F. If EF and there is at least one element in F but not in E, we say that E is a proper subset of F.

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Publisher: Cambridge University Press
Print publication year: 1973

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  • Theory of sets
  • S. J. Taylor
  • Book: Introduction to Measure and Integration
  • Online publication: 21 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662478.002
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  • Theory of sets
  • S. J. Taylor
  • Book: Introduction to Measure and Integration
  • Online publication: 21 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662478.002
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Theory of sets
  • S. J. Taylor
  • Book: Introduction to Measure and Integration
  • Online publication: 21 March 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662478.002
Available formats
×