ISBN-10: 8835946654

ISBN-13: 9788835946656

Similar analysis books

Download e-book for iPad: Neoliberalismus: Analysen und Alternativen by Christoph Butterwegge, Bettina Lösch, Ralf (Eds.) Ptak, Ralf

Der Neoliberalismus hat in den letzten Jahren weite Bereiche unserer Gesellschaft geprägt. Es ist ihm gelungen, zumindest in einem Großteil der medialen Öffe- lichkeit die Legitimität des grundgesetzlich geschützten Sozialstaates – erstmals nach 1945 – zu erschüttern und dessen Säulen ins Wanken zu bringen.

Extra resources for Corso di matematica superiore

Sample text

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 .