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Introduction to discrete structures. This is an introduction to some of the discrete structures that are used in computability. The discrete structures used are Peano Arithmetic, which we will deal with in this chapter. The foundation of computability is recursive functions. In this chapter we will see some basic principles of recursion and also how to do simple recursive functions using basic operators. We then use the primitive recursive functions to define our first recursive structures. We also give some examples and finally review Peano Arithmetic. Discrete Structures. As we have seen in the previous chapter, there is a big difference between algebra and arithmetic. A group can be embedded in an algebra of a field and thus become an algebraic object. In the case of numbers, however, there is no obvious set of axioms that we can use to get the two notions to match. One of the main differences between arithmetic and algebra is the notion of division. In arithmetic, if x and y are natural numbers, then there is a natural number n such that xn = y. In algebra, division is not always possible. The simplest example is the field of complex numbers, where x and y can be multiplied together to yield a result that can be quite far from x and y. Real numbers are a special type of algebraic number which can be described as a sequence of complex numbers. Since real numbers are not algebraic, they cannot be described as elements of an algebra. We now study real numbers. We shall first deal with positive reals. The reals are a subset of the positive reals. The positive reals are the set of all positive real numbers. Any real number can be written as a fraction in lowest terms, since a real number must have a decimal representation which can be written in terms of digits of a decimal expansion.. This is a discrete version of a function. A function f: A -> B is defined as a relation A B. The relation A B can be a subset of A x B, in which case we say that f is defined on the set A. One of the most important relations used to define functions are the domain and range relations.
Download or read online a manual to discretes structures logic and computability (in french) with other manuals, textbooks, and other educational resources. Gedatteerde ontwikkelingswerkwijze en technische informatie. En ingetrokken internet-informatie en notitie van de Stichting Odra.
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Feb 16, 2015. It is also an introduction to the Discrete Structures used in Computability. This book is full of good exercises for the reader to practice the concepts they have learned in this book.
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