# Is this predicate logic true?

∃ x ∈ R, ∀ y ∈ R x ≥ y

Write the statement in English. A complete answer will not use any mathematical notation, nor the symbols x and y. Write down the truth value of the statement. Write down the negation of the statement in symbols and in English.

my soln:

some real numbers are greater or equal to all real numbers. false noreal numbers are greater or equal to all real numbers

Pls help

• Your version is fine. More idiomatic, in English, would be something like "There is a largest real number." – André Nicolas Mar 20 '14 at 2:52
• You are almost correct, although I would say "some real number is greater or equal to all real numbers." Or more precisely, "there is/exists a real number..." – 4ae1e1 Mar 20 '14 at 2:52
• The statement translates as: There exists some other real number for all real numbers such that the first is greater than or equal to the latter The statement is false. – user122283 Mar 20 '14 at 3:12
• what about the negation? is mine ok? – logicunix Mar 20 '14 at 3:38
• Might as well write the formal negation as well and then see if your sentence corresponds to it. The negation is, $\neg(\exists x\in\mathbb R, \forall y\in \mathbb R, x\geq y)$. Recall that $\neg(\exists x,\phi(x))=\forall x,(\neg\phi(x))$ and $\neg(\forall x,\phi(x))=\exists x,(\neg \phi(x)$. Then the above negation becomes $\forall x\in \mathbb R, \exists y\in\mathbb R, x< y$. Is it true that if someone hands you a real number, you can give a real number greater than it? – Rachmaninoff Mar 20 '14 at 3:51

## 2 Answers

There exists a real number $x$, such that, for every real number $y$, $x$ is either greater than $y$ or equal to $y$. In other words, there is a real number which is greater than or equal to all real numbers.

This statement is false of course.

Proof by contradiction:

• Assume that there is such number $x$.
• Observe the number $y=x+1$.
• Obviously, $x$ is neither greater than $y$ nor equal to $y$.

$\exists(x \in \mathbb{R})[\forall (y \in \mathbb{R} ) [x \geq y]]$

It means:

There exists a number in reals for which all real numbers are smaller or equal to it.

It's $\infty$, of course.

• $\infty$ is not a real number. Saying "It's $\infty$" is not accurate. – N8tron May 25 '14 at 12:32