Abstract
The patterns of intersection-enclosure and generation are defined. They are applied to show that any subset of an additive subgroup of \(R\) is contained in a smallest subgroup. Intersection enclosure is applied to sigma algebras, convex hulls, and linear subspaces. It is shown that the smallest subfield of \(R\) containing the rationals and \(\sqrt{2}\) is not \(R\).
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Notes
- 1.
In fact, \(x^3-m\) only has one real root and an alternate route to proving this theorem is to show that.
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© 2015 Springer International Publishing Switzerland
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Joshi, M. (2015). Intersection-Enclosure and Generation. In: Proof Patterns. Springer, Cham. https://doi.org/10.1007/978-3-319-16250-8_6
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DOI: https://doi.org/10.1007/978-3-319-16250-8_6
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16249-2
Online ISBN: 978-3-319-16250-8
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