Is Every Rational Number Always a Ratio of Two Integers?

[kuahji]

Rewrite the following statement formally. Use variables and include both quantifiers \forall and \exists in your answer.

Statement: Every rational number can be written as a ratio of some two integers.

If I didn't have to use \exists I'd write it as follows

\forallrational numbers x, x is a ratio of two integers.

But I can't think of a way or any reason why I'd want to include the quantifier \exists.

 

[Rainbow Child]

Rewrite the following statement formally. Use variables and include both quantifiers \forall and \exists in your answer.

Statement: Every rational number can be written as a ratio of some two integers.

If I didn't have to use \exists I'd write it as follows

\forallrational numbers x, x is a ratio of two integers.

But I can't think of a way or any reason why I'd want to include the quantifier \exists.

You must use variables in your answer.

 

[HallsofIvy]

What does "x is a ratio of two integers" mean? That's where you need \exists.

 

[kuahji]

Ok thanks, I rewrote it as
\forall rational numbers x, \exists a rational number y and a rational number z such that x=y/z.

One more question if anyone has time to help me with.

Rewrite the statement formally.
Statement: There is a program that gives the correct answer to every question that is posed to it.

So I rewrote it as
\exists a program p such that \forallquestions q, p always answers q correctly.
Is this incorrectly because I have "correctly" as the final word?
The book shows the answer as
\exists a program P such that \forall questions Q posed to P, P gives the correct answer to Q.

I didn't know if these was some technicality that would make my answer incorrect as apposed to the book's answer.

 

[Mystic998]

It's mostly right except the original statement is that any rational number can be written as a ratio of 2 integers.

For the second one, the only real problem I see is that "always" is redundant.