Abstract
This chapter starts with the counting or natural numbers, formalising them using structured types, thus allowing the definition of the standard numeric operators such as addition and multiplication. Using the natural numbers, the notions of mathematical and strong induction are formalised and illustrated through examples. Other classes of numbers, such as the integers, rational numbers and real numbers are also discussed. The notion of cardinality is presented, showing how one can reason about the size of finite, but also infinite sets.
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Notes
- 1.
Theoretical, because we do not limit the answer depending on the size of your kitchen or the number of angels or sheep that exist in the universe.
- 2.
As the complexity of the proofs in the rest of this chapter increases, we will not be giving full rigorous proofs, since these would be extremely long and tedious to follow. Instead, we will be giving proof sketches, which indicate the outline of the structure of a rigorous proof if we were to write one.
- 3.
RSA stands for Rivest, Shamir and Adleman, the names of the scientists who published this algorithm.
- 4.
Even in myths and legends in which someone is assigned a seemingly never-ending task, it usually consists of counting a finite, even if very big, collection—counting the grains of sand on a beach, or the number of drops of water in an ocean, as opposed to counting infinite collections of objects, such as counting all the numbers one can think of.
- 5.
The special case when f′(k+1)=m has to be treated differently, and is left as an exercise.
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© 2012 Springer-Verlag Berlin Heidelberg
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Pace, G.J. (2012). Numbers. In: Mathematics of Discrete Structures for Computer Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29840-0_9
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DOI: https://doi.org/10.1007/978-3-642-29840-0_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29839-4
Online ISBN: 978-3-642-29840-0
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