MATH 245 CH3

# MATH 245 CH3 - DISCRETE MATHEMATICS W W L CHEN c W W L Chen...

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DISCRETE MATHEMATICS W W L CHEN c ± W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 3 THE NATURAL NUMBERS 3.1. Introduction The set of natural numbers is usually given by N = { 1 , 2 , 3 ,... } . However, this deﬁnition does not bring out some of the main properties of the set N in a natural way. The following more complicated deﬁnition is therefore sometimes preferred. Definition. The set N of all natural numbers is deﬁned by the following four conditions: (N1) 1 N . (N2) If n N , then the number n + 1, called the successor of n , also belongs to N . (N3) Every n N other than 1 is the successor of some number in N . (WO) Every non-empty subset of N has a least element. The condition (WO) is called the Well-ordering principle. To explain the signiﬁcance of each of these four requirements, note that the conditions (N1) and (N2) together imply that N contains 1 , 2 , 3 ,.... However, these two conditions alone are insuﬃcient to exclude from N numbers such as 5 . 5. Now, if N contained 5 . 5, then by condition (N3), N must also contain 4 . 5 , 3 . 5 , 2 . 5 , 1 . 5 , 0 . 5 , - 0 . 5 , - 1 . 5 , - 2 . 5 ,..., and so would not have a least element. We therefore exclude this possibility by stipulating that N has a least element. This is achieved by the condition (WO). Chapter 3 : The Natural Numbers page 1 of 6

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Discrete Mathematics c ± W W L Chen, 1982, 2008 3.2. Induction It can be shown that the condition (WO) implies the Principle of induction. The following two forms of the Principle of induction are particularly useful. PRINCIPLE OF INDUCTION (WEAK FORM). Suppose that the statement p ( . ) satisﬁes the following conditions: (PIW1) p (1) is true; and (PIW2) p ( n + 1) is true whenever p ( n ) is true.
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MATH 245 CH3 - DISCRETE MATHEMATICS W W L CHEN c W W L Chen...

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