Is this predicate logic true?

Question

∃ 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

Answer 1

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

It means:

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

It's $\infty$, of course.

Answer 2

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 $y$.