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A/ C 1, we say that b is an immediate extension of a. If u D u1 u2 um is a string of length m and v D v1 v2 vn is a string of length n, we can form their concatenation, denoted by u v, to be the string of length mCn obtained by “writing u followed by v,” as in: u v WD u1 u2 u m v1 v2 vn : Thus u is a prefix of w if and only if w D u v for some string v. v/. Let A be an alphabet. u/ C 1. It is often useful to regard an infinite sequence a 2 AN as an infinite string a D a1 a2 an of letters from the alphabet A.

For any m; n: n mCn m C D ; 1 1 1 m n m n D ; 1 1 1 and m n < if and only if m < n: 1 1 Problem 126. n/ nÁ • S is interpreted as . 1 1 At this point, the natural numbers and the integral ratios become interchangeable since all the properties of the natural numbers listed in the initial sections are possessed by the integral ratios. Therefore, we throw away the natural numbers3 and use the corresponding integral ratios in their place. The old natural numbers are not used directly anymore, and so we now deal with only one type of numbers, namely the ratios, which include the “new natural numbers” (really the integral ratios) as a subset.

Problem 63. n 6< n for all n; that is, there are no n; p such that n D n C p. Problem 64. For all n, either 1 < n or 1 D n. Also, there is no n with n < 1. Thus 1 is “the least natural number” (less than all other natural numbers). Problem 65. m/ D n. Problem 66. m C k < n C k if and only if m < n. Recall from the previous chapter that a relation on a set is called a linear order if it is transitive, irreflexive, and connected on the set. Theorem 67. <, as defined above, is a linear order on the natural numbers.