Download e-book for iPad: Corso di matematica superiore by Vladimir Ivanovic Smirnov

By Vladimir Ivanovic Smirnov

ISBN-10: 8835946654

ISBN-13: 9788835946656

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PROPERTIES OF NATURAL NUMBERS 43 The proof of the next theorem provides several other, somewhat more complicated examples of inductive proofs. 2 Theorem ( N , <) is a linearly ordered set. Pm0f. (i) The relation < is transitive on N. We have to prove that for all k,m,n E N , k < m and m < n imply k < n. , we use as P ( x ) the property "for all k , m € N, i f k < m a n d m < X , then k < x . " (a) P ( 0 ) holds. P(0) asserts: for all k, m E N, if k < m and m < 0, then k < 0. l(i), there is no m E N such that m < 0, so P(0) is trivially true.

So b is the greatest element of t , h s~r t of all lower bounds of B. ( c ) This is obvious. (d) This is only a reformulation of the definition of infimum. he suprcrrlu~ll of B, if they exist. If B is linearly ordered, we also use ~ n i n ( B )and rnax(B) t ( ~ denote the minimal ([east) and the maximal (greatest) elements of B, if the). exist . 16 Example Let 5 be the usual ordering of the set of real numbers; let, B1 = { X 1 0 < X < I ) , B2 = { X 1 0 5 X < l } , B3 = { X ( z > 0 ) : anti B4 = { X 1 X < 0).

Therefore, a: belongs to the inverse image of B under R if and only if for some y E B, (y, X) E R- '. e.. U if and only if a: belongs to the image of B under R-'. In the rest of the book we often define various relations. e.. sets of ordered pairs having some particular property. To simplify our notation, we introduce the following conventions. Instead of {W ) W = (a:, y) for some X and y such that P ( x , y ) ) , we simply write Y) l P(x,Y)). For example, the inverse of R could be described in this notation as { ( X .

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Corso di matematica superiore by Vladimir Ivanovic Smirnov

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