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I need help with a set problem

Given:

$$A=\{(\sqrt{n}+2) \in \Bbb Z \ /\ \ 16\le n^2 \le 1296 \}$$

$$B=\{({3m-2}) \in A \ /\ \ 4 \le 4m+3 \le 17 \}$$

Calculate the value of : $$n(A)\times n(B)$$

So far I've got into $$ A = \{-8;-7;-6;-5;-4;4;5;6;7;8\} $$ $$B= \{4;5;6;7;8\}$$

therefore $$n(A) = 10$$ $$n(B) = 5$$ However I don't know if this is correct as the result does not match any of the options given as answer.

Harry
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2 Answers2

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Not quite: The symbol $\sqrt{n}$ usually refers to the nonnegative square root of $n$, so $A$ should only contain positive numbers. Now

$$\sqrt{1296} = \sqrt{6^4} = 36$$

so the smallest element of $A$ corresponds to $\sqrt{4} + 2$, while the largest corresponds to $\sqrt{36} + 2$; that is, it's equivalent to write

$$A = \{\sqrt{n} + 2 : 4 \le n \le 36\}$$

  • wouldn't it be the same to write $$ A = { \sqrt{n}+2 : 4 \le \sqrt{n}+2 \le 8 } $$ ? – Harry Mar 08 '14 at 03:18
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    No, that is not the same because you don't restrict $\sqrt n$ to naturals so $2+\sqrt 5$ would be included. You are thinking in the right direction. – Ross Millikan Mar 08 '14 at 18:09
  • Okay so now after correcting I got $$ A={4;5;6;7;8} \ & \ B = {4;5;6;7;8}$$ but the result of n(a)x n(b) still differs from the given options, Maybe i'm still doing it wrong? Thank you in Advance – Harry Mar 09 '14 at 05:01
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Hint. The symbol $\sqrt{\ }$ means the positive square root. It doesn't mean plus- or- minus unless you actually write $\pm$.

David
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