Truth Statement of a domain

Question

$\exists \ x \ \forall \ y\ (x\leq y^2)$

$\mathbb{N}:$ True

$\mathbb{Z}:$ True

$\mathbb{R}^+:$ False

I understand why it is true with natural numbers and integers, but why is it false for the set of positive real numbers? What is an example of $(x\leq y^2)$ for $\mathbb{R}^+$?

score 3 · Accepted Answer · answered Jun 13 '17 at 15:42

3

There is no such real positive number that it is smaller than the square of any positive real number you pick, since the squares get arbitrarily small.

For the natural numbers it would be 0 or 1, depends on whether you include 0. For integers it would be zero.

answered Jun 13 '17 at 15:42

B.Swan

2,469

What if X=Pi and y=Pi^2? wouldn't x≤y^2? – QuestionQuestion Jun 13 '17 at 16:02
1

Yes, however the statement is that your x is smaller than the square of any y you pick. It is true for the y you picked, but is $\pi$ smaller than 1²? Clearly not, – B.Swan Jun 13 '17 at 16:05
OOOHH. I see now. Thank you for taking the time to clarify. – QuestionQuestion Jun 13 '17 at 16:06

Truth Statement of a domain

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