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**Extra resources for ANALYSE, CHAPITRE 6: APPLICATION DE L'INTÉGRALE DÉFINIE**

**Example text**

One of the basic questions that arises is this: when is the Cartesian product nonempty? Obviously, if a t least one set Yk = 0, then Y, = 0. But if all Y, # 0, is the Cartesian product (i) ,EX nonempty necessarily? Although the answer may seem obvious, we must turn to the axiom of choice. In other words, the Cartesian product of a family of sets is nonempty if and only if there exists a t least one choice function for this family. (ii) We said that the Cartesian product of the family of sets {Y, : x E X} is the collection of all functions from X to Y,, x E X.

Supplementary Historical Note. Modern set theory was founded by Georg Cantor, in a sequence of several articles that appeared between 1870 and 1880. One of these articles, [iber eine Eigenschafl des Inbegrifles allen reellen algebraischen Zahlen, appeared in Crelle's Journal in 1874, and is said to have given birth to set theory. Georg Cantor was born of Danish parents (both of Jewish descents) in St. Petersburg, Russia, in 1845, and lived there until 1856, when his parents moved to Frankfurt, Germany.

A set A is called at most countable if I A I (vii) An infinite set, which is not countable is called uncountable. (viii) A set A is said to have the cardinalzty of continuum if it is equipotent with the set R of real numbers and we write I A I = a. 2 Remark. 1). U If sets A and B have only finitely many elements, then A (i) if and only if they have the same number of elements. B B In contrast with finite sets, an infinite set can be equipotent (ii) with a proper subset of itself. Consider A = {1,3,5,.