Proving Mathematical Statements: a Real-Life Example

[Rob Hal]

Hi all,

If I have these two statements given to me, and I have to determine whether they are true or not.

a) $$\forall x \epsilon R$$ $$\exists y \epsilon R$$ $$(y^2 = x^2 + 1)$$
b) $$\exists y \epsilon R$$ $$\forall x \epsilon R$$ $$(y^2 = x^2 + 1)$$

Now, to me, they both mean exactly the same thing, and both can be shown to be false by setting x = 2, then y is not a real number.

However, seeing that the question specifically asks to prove just those two statements, I'm wondering if perhaps I am interpreting them wrong and they actually mean two different things.

Thanks in advance for any advice,
Robbie

 

[TD]

Now, to me, they both mean exactly the same thing, and both can be shown to be false by setting x = 2, then y is not a real number.

If x = 2 then $y^2 = 2^2 + 1 \Leftrightarrow y = \pm \sqrt 5$. Those are real numbers, no?

 

[Rob Hal]

lol... yeah...
I was thinking I was looking for rationals only... whoops...

Still, is there any difference in the two statements themselves?

 

[honestrosewater]

Try a simpler one to see how the order of the quantifiers makes a difference:

$$\forall x \in \mathbb{R} \ \exists y \in \mathbb{R} \ (x = y)$$

This says that for every real x that I choose, I can find at least one real y that is equal to that x. This is obviously true, since x = x.

$$\exists y \in \mathbb{R} \ \forall x \in \mathbb{R} \ (x = y)$$

This says that I can find at least one real y that is equal to every real x. Well, there's more than one real number, so this is false.