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sols22 - 22. NOVEMBER 22ND: CARDINALITY IV 51 22. November...

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22. NOVEMBER 22ND: CARDINALITY IV 51 22. November 22nd: Cardinality IV 22.1 . Prove that if S is uncountable, and C S is countable, then | S ° C | = | S | . Answer: Let N ( S ° C )beaninFn itecountab leset ,andlet C ° = C N .The union of two countable sets is countable (Example 21.7), so C ° is inFnite countable: C ° = N .S inc ew ehav e S =( S ° C ° ) C ° and S ° C =( S ° C ° ) N it follows that | S | = | S ° C | , by Lemma 20.1. ° 22.2 . Prove that the set of transcendental number has the cardinality of the con- tinuum. Answer: This is an immediate application of Exercise 22.1: Take S = R ,and C = A ,theseto fa lgebra icnumbers ;thens ince A is countable, the result of Exercise 22.1 shows that the set R°A of transcendental numbers has the same cardinality as R . ° 22.3 . Prove that the set C of complex numbers has the cardinality of the contin- uum. Answer: Complex numbers are numbers of the form a + bi ,where a and b are real numbers, and i is a square root of 1. Thus, C = R × R :theknow ledgeo facomp lex number is equivalent to the knowledge of its real part a and its imaginary part b .I t follows that | C
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.

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sols22 - 22. NOVEMBER 22ND: CARDINALITY IV 51 22. November...

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