By Andranick S. Tanguiane

ISBN-10: 3642765165

ISBN-13: 9783642765162

ISBN-10: 3642765181

ISBN-13: 9783642765186

Aggregation is the conjunction of knowledge, aimed toward its compact represen tation. Any time while the totality of knowledge is defined by way of basic ized signs, traditional counts, usual representatives and attribute dependences, one without delay or not directly offers with aggregation. It comprises revealing the main major features and exact positive factors, quanti tative and qualitative research. for this reason, the data turns into adaptable for extra processing and handy for human notion. Aggregation is known in economics, records, administration, making plans, procedure research, and plenty of different fields. because of this aggregation is so very important in info professional cessing. Aggregation of personal tastes is a selected case of the overall challenge of ag gregation. It arises in multicriteria decision-making and collective selection, while a suite of possible choices should be ordered with appreciate to contradicting standards, or numerous person critiques. even if, regardless of obvious similarity the issues of multicriteria decision-making and collective selection are a bit assorted. certainly, an development in a few requirements on the rate of irritate ing others isn't the similar because the pride of pursuits of a few participants to the bias of the remaining. within the former case the reciprocal compensations are thought of inside a definite entirety; within the latter we infringe upon the rights of self sufficient contributors. in addition, in multicriteria decision-making one usu best friend takes under consideration aim components, while in collective selection one has to check subjective evaluations which can't be measured properly.

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**Example text**

The lexicographic order cannot be represented by a goal function, otherwise for every x it would be f(x,O) < f(x, 1), what implies the existence of an uncountable family of disjoint intervals on the real line. Since each interval contains a rational number, it is impossible. 14 every weak order P on X separates X into disjoint classes of indifferent elements. Obviously, these classes are strictly ordered. Therefore we can reduce the problem of numerical representation of a weak order to the problem of numerical representation of the associated strictly ordered quotient set.

E. a reflexive, symmetric, and transitive binary relation. 2 Binary Relations and Orderings 29 PROOF. Let P be a weak order on X. 13 the relation P is transitive, whence P is a partial order. Let us show that the indifference", defined with respect to P is an equivalence. y '" x, which proves that the indifference is symmetric. Since P is asymmetric, P is irreflexive, and (x, x) ¢: P for every x E X, whence x '" x, which proves that the indifference is reflexive. Finally, if x '" y and y '" z for some x, y, z E X, then we have (x,y) ¢: P, (y,x) ¢: P, (y,z) ¢: P, (z,y) ¢: P.

Since the image f(A) of every interval A c X is an interval in Y, it follows that f-1(s) E A for s E 5 n f(A). If X had an uncountable set of jumps, then the monotone mapping f(x) would transfer it into an uncountable set of disjoint real intervals, which is impossible (d. 2). We show that condition 3 implies condition 4. The existence of a countable dense subset in X follows from the count ability of the base. We shall prove that the set of jumps is countable. For each jump, which is an empty interval (a; b) = {x : x E X, a -< x, x -< b}, define an open neighborhood of point a to be U(a) = {x : x E X, x -< b}.

### Aggregation and Representation of Preferences: Introduction to Mathematical Theory of Democracy by Andranick S. Tanguiane

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