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1.1.1: Extensionality
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When we consider sets, we don’t care about the order of their elements, or how many times they are specified.
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1.1.2: Subsets and Power Sets
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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 \(A\).
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1.1.3: Some Important Sets
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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.
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1.1.4: Unions and Intersections
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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.
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1.1.5: Pairs, Tuples, Cartesian Products
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It follows from extensionality that sets have no order to their elements. So if we want to represent order, we use ordered pairs \(\langle x, y \rangle\).
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1.1.6: Russell’s Paradox
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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.
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1.1.7: Summary
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