lect_40 - Cardinality Margaret M. Fleck 5 May 2010 This is...

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Unformatted text preview: Cardinality Margaret M. Fleck 5 May 2010 This is a half lecture due to the makeup quiz. It discussed cardinality, an interesting topic but which doesnt have an obvious fixed place in the syllabus. Its covered at the very end of section 2.4 in Rosen. 1 The rationals and the reals Youre familiar with three basic sets of numbers: the integers, the rationals, and the reals. The integers are obviously discrete, in that theres a big gap between successive pairs of integers. To a first approximation, the rational numbers and the real numbers seem pretty similar. The rationals are dense in the reals: if I pick any real number x and a distance , there is always a rational number within distance of x . Between any two real numbers, there is always a rational number. We know that the reals and the rationals are different sets, because weve shown that a few special numbers are not rational, e.g. and 2. However, these irrational numbers seem like isolated cases. In fact, this intuition is entirely wrong: the vast majority of real numbers are irrational and the rationals are quite a small subset of the reals....
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This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.

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lect_40 - Cardinality Margaret M. Fleck 5 May 2010 This is...

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