Question

I don't understand how you know the truth value of each statement if the universe of each variable consist of all real numbers?

Answer 1

\[\forall x \forall y \exists z (x^2+y^2=z^2)\]

Answer 2

I don't know how to do it without just randomly plugging numbers in

Answer 3

Assuming all of the usual properties of real numbers, then:

1. For any arbitrary x and y\[x^2 + y^2 \]
is itself a real number greater than or equal to zero.

2. For any real number a say such that
\[a \geq 0\]
there is a real number z such that
\[z^2 = a\]

3. Now combining 1 and 2, given any arbitrary real numbers x and y there is a real number z such that
\[x^2 + y^2 = z^2\]

*****

Ok, now you might say, "I concede that 1. and 2. imply 3. But how do you know 1. and 2. are true?"

Well, if you want to be logically thorough, you need to prove them starting with the axioms of the real number system. It turns out that using the usual axioms it is indeed the case that 1. and 2. are true, but if you don't believe it, you'll have to show it for yourself.

Answer 4

Thank you! You explained a lot better than my prof