1.1.7: Summary — Read — Invisible Network

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A set is a collection of objects, the elements of the set. We write \(x \in A\) if \(x\) is an element of \(A\). Sets are extensional—they are completely determined by their elements. Sets are specified by listing the elements explicitly or by giving a property the elements share (abstraction). Extensionality means that the order or way of listing or specifying the elements of a set doesn’t matter. To prove that \(A\) and \(B\) are the same set (\(A = B\)) one has to prove that every element of \(X\) is an element of \(Y\) and vice versa.

Important sets include the natural (\(\Nat\)), integer (\(\Int\)), rational (\(\Rat\)), and real (\(\Real\)) numbers, but also strings (\(X^*\)) and infinite sequences (\(X^\omega\)) of objects. \(A\) is a subset of \(B\), \(A \subseteq B\), if every element of \(A\) is also one of \(B\). The collection of all subsets of a set \(B\) is itself a set, the power set \(\Pow{B}\) of \(B\). We can form the union \(A \cup B\) and intersection \(A \cap B\) of sets. An ordered pair \(\tuple{x, y}\) consists of two objects \(x\) and \(y\), but in that specific order. The pairs \(\tuple{x, y}\) and \(\tuple{y, x}\) are different pairs (unless \(x = y\)). The set of all pairs \(\tuple{x, y}\) where \(x \in A\) and \(y \in B\) is called the Cartesian product \(A \times B\) of \(A\) and \(B\). We write \(A^2\) for \(A \times A\); so for instance \(\Nat^2\) is the set of pairs of natural numbers.

This page titled 1.1.7: Summary was last modified on Fri, 08 Mar 2024 05:56:22 GMT and is shared under a CC BY license and was authored, remixed, and/or curated by Richard Zach et al. (Open Logic Project) .

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