Abstract
An existence statement asserts that there is a mathematical quantity that has a stated property. We distinguish between constructive and non-constructive existence proofs: the former exhibit the object in question explicitly, the latter do not. We examine definitions as instances of existence statements, and recursive definitions as instances of induction. Common mistakes found in definitions are considered in the last section and in the exercises.
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Notes
- 1.
There was no guess here: first I chose \(x_+\), and then I derived the equation of which \(x_+\) is a solution.
- 2.
Some authors use ‘effective’ to mean ‘constructive’.
- 3.
For \(n>1\), the integer \(1+n!\) is followed by \(n-1\) composite integers. (Why?)
- 4.
Conjectured by Joseph Bertrand (French: 1822–1900) in 1845. Proved by Pafnuty Chebyshev (Russian: 1821–1894) in 1850.
- 5.
Leonardo Pisano, known as Fibonacci (Italian: 1170–ca.1240).
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© 2014 Springer-Verlag London
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Vivaldi, F. (2014). Existence and Definitions. In: Mathematical Writing. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-6527-9_9
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DOI: https://doi.org/10.1007/978-1-4471-6527-9_9
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Publisher Name: Springer, London
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Online ISBN: 978-1-4471-6527-9
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