# Thread: Need help writing the logical formulation of statements

1. ## Need help writing the logical formulation of statements

I am needing some help writing the logical formulation of statements.
For example, here are some statements:
1. Between any two real numbers there is at least one rational number and one irrational number.
2. If r1 and r2 are any real numbers satisfying r1 < r2, then r1+a < r2+a for each real number a.
3. If a, b, and c are any real numbers, then a(b+c)=ab+ac.

I think the problem is that my teacher didn't give us any type of examples of doing this, so I am having a hard time figuring them out. I would appreciate any and ALL advice you can give me on doing these. Also, how do you know what the universe is for these things?

Thank you so much for any help.

2. ## Re: Need help writing the logical formulation of statements

1. $\displaystyle \forall x\in\mathbb{R}\,\forall y\in\mathbb{R}\,\Big(x<y\to\big(\exists u\in\mathbb{Q}\,(x<u\land u<y)\big)\land\big(\exists v\in\mathbb{R}\setminus\mathbb{Q}\,(x<v\land v<y)\big)\Big)$

3. $\displaystyle \forall a\in\mathbb{R}\,\forall b\in\mathbb{R}\,\forall c\in\mathbb{R}\,a(b+c)=ab+ac$

Try writing 2.