1.1: Sets

Sets, Logic, Computation · LibreTexts

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    From LibreTexts, under CC BY, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    • 1.1.1: Extensionality
      When we consider sets, we don’t care about the order of their elements, or how many times they are specified.
    • 1.1.2: Subsets and Power Sets
      If every element of a set \(A\) is also an element of \(B\), then we say that \(A\) is a subset of \(B\). The set consisting of all subsets of a set \(A\) is called the power set of \(A\).
    • 1.1.3: Some Important Sets
      Important sets include the natural (\(\mathbb{N}\)), integer (\(\mathbb{Z}\)), rational (\(\mathbb{Q}\)), and real (\(\mathbb{R}\)) numbers, but also strings (\(X^*\)) and infinite sequences (\(X^\omega\)) of objects.
    • 1.1.4: Unions and Intersections
      The union of two sets \(A\) and \(B\), written \(A \cup B\), is the set of all things which are elements of \(A\), \(B\), or both. The intersection \(A \cap B\) of two sets is the set of elements they have in common.
    • 1.1.5: Pairs, Tuples, Cartesian Products
      It follows from extensionality that sets have no order to their elements. So if we want to represent order, we use ordered pairs \(\langle x, y \rangle\).
    • 1.1.6: Russell’s Paradox
      Some properties do not define sets. If they all did, then we would run into outright contradictions. The most famous example of this is Russell’s Paradox.
    • 1.1.7: Summary

    This page titled 1.1: Sets was last modified on Sun, 12 Sep 2021 15:28:39 GMT and is shared under a CC BY license and was authored, remixed, and/or curated by Richard Zach et al. (Open Logic Project) .