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Unformatted text preview: (ii) Before we decide whether the number is irrational, let us reduce the fraction under the square root: 54 150 = 27 Â· 2 75 Â· 2 = 27 75 = 9 Â· 3 25 Â· 3 = 9 25 . 11 Now we can simplify: r 54 150 = r 9 25 = âˆš 9 âˆš 25 = 3 5 . Thus r 54 150 is rational . (iii) Suppose Q := 3 Ï€ is rational. Then Ï€ = Q/ 3 will be rational which is a contradiction. So 3 Ï€ is irrational . (iv) 2 / 17 is the ratio of two integers, so 2 / 17 is rational . The correct answer is (e) . Solution of problem 1.9: (1) is True. We can construct a onetoone cor respondence all natural numbers ending with 7 o / { all negative integers } explicitly by pairing: a natural number n ending with 7 o / the negative integer ( n 7) / 10 . (2) is False. If we have a onetoone correspondence f between A and B and we have a onetoone correspondence g between A and C , then we can construct a onetoone correspondence between B and C . Indeed, let x be a number in B , then f pairs b with a unique element y in A . On the other hand g will pair y with a unique element z in C . So we can define a correspondence h between B and C by pairing x and z . The correspondence h will be automatically a onetoone correspondence because of the uniquencess of the pairings f and g . 12 (3) is True. A rational numbers whose denominator is a power of 3 can be written uniquely as a reduced fraction m 3 n where n is an integer with n â‰¥ 0, and m is an integer that is not divisble by 3. Such a number is between 0 and 1 if and only if 0 â‰¤ m â‰¤ 3 n . So for instance the numbers whose numerators are all equal to 1 and whose denominators are powers of 3 are in our set. But we can easily construct a onetoone correspondence all natural numbers of the form 1 3 n , n = 1 , 2 , .. . o / { all natural numbers } by pairing 1 / 3 n with n . Therefore the set of all numbers of the form 1 / 3 n is infinite. But our set contains the set of all of the form 1 / 3 n is infinite, so it also must be infinite. Solution of problem 1.10: The sets A and B have the same cardinality: we can match a natural number n with the number that is made out of n digits, each of which is equalto 5. The set C has a bigger cardinality. We can see this by Cantorâ€™s diagonalization argument. Indeed, suppose that we can find a onetoone correspondence between the set C and all natural numbers. Let c n denote the number in C that corresponds to the natural number n . We will construct a number a in C that is not equal to any one of the c n â€™s. Define a to be the number between 0 and 1 for which: nth digit of a after the decimal point = 3 if the nth digit of c n after the decimal point is 5; 5 if the nth digit of c n after the decimal point is 3. This defines a completely and ensures that a and c n have a different n th digit after the decimal place. Therefore a can not be equal to c n for 13 any n . Which is a contradiction since a âˆˆ C and the c n â€™s enumerated all the numbers in C . This shows that C has a bigger cardinality than A (and hence than B ). The correct answer is (c) . 14...
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 Spring '08
 schneps
 Calculus, Decimal, Natural number, Prime number

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