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I'm trying to find the cardinality of the below:

$$A = \left\{ x\in \mathbb{Z}: \bigg|\frac{3x^3 + x^2 - 2x + 4}{3x + 4}\bigg| \geq (2^{50} -1 ) \right\}$$

and the set

$$B = \left\{ x\in \mathbb{Z}: \frac{3x^3 + x^2 - 2x + 4}{3x + 4} = 0 \right\}$$

I'm asked to find the following:

Cadinality of |A|, and of |B| as well as the cardinality of $$ |A\cap B |$$ $$|A \cup B| $$ $$|A \otimes B|$$

I'm kind of looking for where I should start, I believe that |B| should be infinite, since there's not really a "limit" on it, but I'm not really sure how to get started on finding these.

Thanks in advance for any guidance!

dustyblade
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1 Answers1

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$B$ is a good place to start, but I think you'll find instead that $B$ has finite cardinality. You must solve the equation $$3x^3 + x^2 - 2x + 4 = 0$$ This is the equation you'll get after deducing that $3x+4$ cannot equal $0$ since elements of $B$ cannot be non-integers. So it's safe to multiply $\frac{3x^3 + x^2 - 2x + 4}{3x + 4} = 0 $ through by $3x+4$. You should find that this cubic equation will have exactly three solutions. Are any of these integers? If so, they are members of $B$, as your set says all $x$ in $B$ are such that $x\in\mathbb{Z}$. For $A$ notice that $$\frac{3x^3 + x^2 - 2x + 4}{3x + 4} \space \text{~} \space x^2$$ so for sufficiently large $x$, and any $y \geq x$ you will eventually find $$\left|\frac{3y^3 + y^2 - 2y + 4}{3y + 4}\right| >2^{50}-1$$

graydad
  • 14,077