What is the difference between the following propositional sentences?

Question

What is the difference between the following sentences:

1. $(\forall y \in \mathbb{R})(\exists x \in \mathbb{R})(x \leq y)$.

   • For each real number y, there exists a real number x such that x is less than or equal to y.

2. $(\exists x \in \mathbb{R})(\forall y \in \mathbb{R})(x \leq y)$

   • There exists a real number x such that for each real number y, x is less than or equal to y.

While I know what these sentences are in english, I still don't know what the difference is between each sentence. How can I go about discerning the difference between the two?

Answer 1

The sequence of choice matters. In 1) you are given some $y$, then have the option to inspect it and find some $x$ which is smaller. You can choose, e.g., $x=y-1$, that will do.

In 2) you have to show me some $x$ (because 2) is the claim that such an $x$ exists) for which you claim that I cannot find a $y$ which is smaller.

You are now invited to present me such an $x$...I will then have a look and see whether I can find a $y$ which is smaller...

(In particular, 1) is a true statement, while 2) is not).

...you could also say that in 1), the choice of x will depend on that of y, while in 2), it's the other way round.

Answer 2

The first sentence says that every $y$ has a $x$ that's less than it. Let's say for example we have $y=3$. Then we know the statement's obviously true since we can choose $x=1$ so that $x\leq y$. In graphical terms, if you take any point on the number line, the sentence says you can shade in everything to the left of the point as a possible $x$.

But the second statement says that there is a certain number $x$, which is lower than every other $y$. Obviously this can't be right since $\mathbb{R}$ is unbounded. Let's say $x$ was $-4$. Well we can just make $y=-5$ to prove the statement wrong. In graphical terms, this statement says that there's a point at the left end of the number line. Of course, this can't be true since $\infty$ isn't a real number.