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So I know it's true and I understand why it's true but I don't understand how I'm supposed to give the answer, as the answer to this practice question is the following:

enter image description here

How exactly am I supposed to layout the answer for this?

Robert Shore
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tinkertailor
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  • if $x$ is fixed, taking $y=\sqrt[3]{x}$ gives the wished result. – Surb Apr 12 '23 at 07:36
  • The answer is complete. What are you missing? – Anne Bauval Apr 12 '23 at 07:39
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    I think this is a really good question actually. Saying $y$ is the cube root of $x$ is almost tautologous, so the answer is (at best) unsatisfying -- at worst, incorrect [because it assumes the existence of a solution to the problem] – Zim Apr 12 '23 at 10:18
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    @Zim I think you are right, it depends on the axioms we have for $\mathbb{R}$. If we've got the usual completeness axiom we could (for $x>0$) go with $y=\sup{z\in\mathbb{R}\mid z^3<x}$ .... – ancient mathematician Apr 12 '23 at 10:22

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How exactly am I supposed to lay out the answer for this?

Let $a$ be an arbitrary real number.

Then put $y=\sqrt[3]a$ so that $y$ is real and $$a-y^3=0.$$ That is, there exists a real $y$ for which $a-y^3=0.$

As $a$ is arbitrary, hence

  • for each real $x,$ there exists a real $y$ for which $x-y^3=0.$

as required.

This sample solutution can be condensed, or even expanded, as preferred.

ryang
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  • I thought the OP expected the answer in predicate logic – D S Apr 12 '23 at 10:19
  • Sorry, I meant with Predicate logic. The question wants me to use predicate logic to show proof. I can explain it like you did but the answer wants me to use a general argument to show proof as x is universally quantified an actual example for ∃y as it's existentially quantified. – tinkertailor Apr 12 '23 at 10:32
  • @tinkertailor You've ostensibly posted the official solution, and even called it "the answer". Yet you keep suggesting that it's not the answer, so this is quite confusing. Please edit your question to clarify what you actually want, and perhaps give an example of how you mean by "general argument". $\quad$ "the answer wants me to use a general argument to show proof as x is universally quantified an actual example for ∃y as it's existentially quantified." You're not making sense; are you sure you do understand the official solution and my rewrite of it? – ryang Apr 12 '23 at 16:50