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$A:= \{ x^2 | x \in (2,3\sqrt{2}] , x \in \mathbb{R} \textrm{ and } x^2 \in \mathbb{Z}\}$

$B:= \{ x^3 | x \in (-2\sqrt[3]{3},3] , x \in \mathbb{R} \textrm{ and } x^3 \in \mathbb{Z}\}$


s(A) + s(B) = ?

s(A) is number of elements of A set. For example if $\ A = \{a,b,c\} $

s(A) = 3 
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    Thank you Willie :) I hope question is clear, now .2011-03-13

1 Answers 1

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Given $x \in (2, 3\sqrt{2}]$, what can $x^2$ be? It must be

$ 2^2 = 4 < x^2 \leq 18 = (3\sqrt{2})^2 $

So $A$ comprises of all integers ($x^2 \in \mathbb{Z}$) bigger than 4 and less-than-or-equal-to 18. There are 14 of them.


Given $x \in (-2\sqrt[3]{3}, 3]$, you similarly find that

$ -24 < x^3 \leq 27 $

which tells you that there are $27 - (-24) = 51$ elements in $B$


Add them together you get $s(A) + s(B) = 65$.

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    @Eray Alakese: the definitions $A:= \{x^2\ldots$ implies that the elements of$A$are the squares, which are required to be integers.2011-03-13