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

$$\forall$$rational 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

$$\forall$$rational 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 $$\forall$$questions 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.