By George Boole

ISBN-10: 140218364X

ISBN-13: 9781402183645

This Elibron Classics e-book is a facsimile reprint of a 1877 version through Macmillan and Co., London.

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**Extra resources for A treatise on differential equations**

**Example text**

Do not use truth tables! 15. If A1, A2, ... are sets then define 00 H Aj j=1 to be the collection of all functions from the natural numbers N into UAj such that f(j) E Aj. What can you say about the cardinality of the set 00 11Aj j=1 when each Aj has the cardinality of Z? What about when each of the Aj has the cardinality of R? 16. Consider the set S of all real numbers obtained by taking rational powers of rational numbers. Is this set countable or uncountable? 17. A closed subset S of the plane is called convex if whenever a, b E S then the line segment connecting a to b lies in S.

Our example will be the set S of all sequences on the set {O, 1}. In other words, S is the set of all infinite sequences of Os and 1s. To see that S is uncountable, assume the contrary. Then there is a first sequence S1 a second sequence ={ 1}00 si j=', S2=Is 2}0c i i=1 , and so forth. This will be a complete enumeration of all the members of S. But now consider the sequence T = {tj} 1 which we construct as follows: If si=Othen make t1=1;if si=1 then set t1=0; If s2 = 0 then make t2 = 1; if 522 = 1 then set t2 = 0; If 93 = 0 then make t3 = 1; if s3 = 1 then set t3 = 0; If s = 0 then make t3 = 1; if s1 = 1 then make tj = 0; etc.

Explain why the statement is true for x = -4. 1. Do not use truth tables! 15. If A1, A2, ... are sets then define 00 H Aj j=1 to be the collection of all functions from the natural numbers N into UAj such that f(j) E Aj. What can you say about the cardinality of the set 00 11Aj j=1 when each Aj has the cardinality of Z? What about when each of the Aj has the cardinality of R? 16. Consider the set S of all real numbers obtained by taking rational powers of rational numbers. Is this set countable or uncountable?

### A treatise on differential equations by George Boole

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