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My assignment is to translate mathematical statements into formula of predicate logic. But before I can write formulas about these statements, I'm really confused about their meaning.

Given that the mathematical notation of a quadratic polynomial with leading coefficient 1 is $P(x) = a_0 + a_1x + x^2$

(1) " $w$ is a root of infinitely many quadratic polynomials with leading coefficient 1 " ==> if I want to write this into a logical statement, should my formula says something to the negation of the statement ? i.e. "there are infinitely many quadratic polynomials with leading coefficient 1 where $w$ is not a root" ?

Or should I say : " For all the coefficients $a_0, a_1$ such that $P(x) = x^2 + a_1x + a_0$ is $0$ when I plug in $x$, then I can always find the other two coefficients $b_0, b_1$ (different than $a_0, a_1$) such that when I replace them with the $a_0, a_1, P(x)$ is still $0$ at $x$" ?

What does the statement actually mean?

(2) " $w$ is a root of polynomials of arbitrarily large degree with leading coefficient 1." ==> Does the statement mean "there are infinitely many polynomials of degree $n$ with leading coefficient 1, such that $w$ is a root ?" or it just means that "if I can find a polynomial of degree $n$ such that $w$ is a root, then I can find a different polynomial of degree $n$ where $w$ is also a root" ?

Would someone please help me on this? I kept thinking about them for couple days, and I'm still lost >_< Thank you in advance :)

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2 Answers 2

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For (1), you know what a quadratic polynomial with leading coefficient 1 is: it's $P(x)=x^2+a_1x+a_0$, so each such polynomial is completely described by giving values for $a_1\text{ and }a_0$. (A polynomial with leading coefficient $1$ is called a "monic" polynomial, by the way.) You also know what it means for a number $w$ to be a root of a polynomial $P(x)$: it means that $P(w) = 0$, that is, when you plug $w$ into the polynomial the result will evaluate to $0$. Now combine what you already know into a rigorous translation of your expression.

For any number $w$, there are infinitely many pairs of numbers $b_1, b_0$ for which $w^2+b_1w+b_0 = 0$.

The only tricky part of this expression is the first clause. The original sentence didn't mention that, but since the sentence didn't say anything else about $w$ the most natural interpretation is that the sentence was intended to apply to *any*n number $w$.

There's one more detail I glossed over, which can be very important: I spoke about "numbers" and that should really be made specific. A reasonable interpretation would be real numbers, but the problem could be read as just talking about integers, for example. The most general choice would be the reals, so in the above definition, preface the two instances of "number" by "real".

For (2), you need to decipher "polynomial of arbitrarily large degree". You appear to know that the degree of a polynomial is the largest integer $k$ that appears in any term $a_kx^k$ of the polynomial, so in this sentence, "arbitrarily large degree" means that you can find a polynomial of degree at least $n$ for any integer $n\ge 0$. Putting all these definitions together you'll have something that expresses

  • For any real number $w$,
  • and any integer $n\ge 0$,
  • there's a monic polynomial $P(x)$ of degree $\ge n$,
  • for which $w$ is a root.

I'll leave the rest for you. Good luck!

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(1) "if I want to write this into a logical statement, should my formula says something to the negation of the statement ?" It doesn't look like the problem says anything about negation, so you shouldn't be thinking along those lines. "i.e. "there are infinitely many quadratic polynomials with leading coefficient 1 where $w$ is not a root" ?" That's actually not the negation of the statement, but as I said, the negation of the statement may not be relevant to the problem you were asked. "Or should I say..." That's closer, but has a serious problem: it only implies there are at least two polynomials, rather than infinitely many. (Also, it's in English, rather than a formula of predicate logic, but I assume you were trying to clarify the meaning first.) Have you discussed any mathematical statements with "infinitely many" in this course before? If not, are there any examples like that in your textbook?

"What does the statement actually mean?" It means there's an infinite bunch of polynomials with leading coefficient 1 that have $w$ as a root. For example, the statement would be true for $w=0$ since $w$ is a root of $x^2$, $x^2+x$, $x^2+2x$, $x^2+3x$,...

(2) "Does the statement mean..." Neither of those. "Arbitrarily large" means "for any size you think of, there's one bigger than that". So it would be true for $w=0$ since $w$ is a root of $x$ and $x^2$ and $x^4$ and $x^8$ and... Whatever natural you say (e.g. 1000), I can name a degree such that $w$ is a root of at least one polynomial (with leading coefficient 1) of that degree (e.g. 1024: $w$ is a root of $x^{1024}+5x$.

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