By Claude Brezinski

Ebook via Brezinski, Claude

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**Sample text**

Proof. Á that is, we show that ²³ i ²)³ and i ²) ³ are disjoint and ²³ i ²)³ complements i ²) ³ up to ? We start with () Suppose i ²)³ and i ²) ³ have a common point %. Then there is & ) such that ²%³ ~ & and & ) such that ²%³ ~ & . Thus, & £ & and is not a single-valued function. 1 24 CHAPTER 1. SET-THEORETIC AND ALGEBRAIC PRELIMINARIES () If i ²)³ does not complement i () ) up to ? Á there will be at least one point % that does not belong to either of these sets ²for they are disjoint as shown above³.

Define ´µ ~ ¸ {¢ ²mod ³¹ ² {³. In other words, ´µ ~ ¸ {¢ E {¢ ~ b ¹. Then any two integers and are related in terms of [ h µ if and only if ´µ . This is an equivalence relation. ³ (#) Let ? be a nonempty set and R ? d ? be a binary relation. ¹ we have with (? Á +) the “smallest” (by the contents of elements of ? d ? ) equivalence relation on ? , where each element forms a singletonclass, and + partitions ? into the “number” of classes corresponding to the quantity of all elements of ?

5 Corollary. Á @ Á ] be a function and let E denote its equivalence kernel. Then ¢ there is a unique reducer function [? Á ? |E Á E ] and [? |E Á @ Á ]Â if is surjectiveÁ the reducer is bijective. 6 Example. Á cos!. # is the quotient set of ? Ecos ~ " f % b ¢ { # 34 CHAPTER 1. SET-THEORETIC AND ALGEBRAIC PRELIMINARIES ~ cosi "cos%# making sÁ sOEcos Á Ecos ! the projection of ? ~ s on its quotient by Ecos . Á cos! 5, it is bijective. 5, cos k Ecos % ~ cos f % b ¢ { ~ cos%À Now we turn to a discussion on the partial order relation and all relevant notions and theorems, which we apply throughout the rest of the book.