### Exercises

1. Show that $-1<0\$
2. Show that $\forall x,y\in\mathbb{Z},\ -(xy)=(-x)y=x(-y)$
3. Show that $\forall z\in\mathbb{Z},\ z^2\ge0$
4. Show that $\forall z\in\mathbb{Z},\ z\ge0\Leftrightarrow-z\le0$
5. Show that $\forall x,y,z\in\mathbb{Z},\ (x<0)\cap(yxz)$
6. Let $p$ be any prime. Show that $\sqrt{p}$ is irrational.
7. Complete the proofs of the simple results given above.
8. Show that the complex numbers $\mathbb C$ cannot be made into an ordered field.
9. Complete the proof of the square roots theorem by giving details for the case $x<1$.
10. Suppose A is a non-empty set of real numbers that is bounded above and let s = sup A. Show that if s is not in A, then for any ε > 0, there exists an element a in A such that s − ε < a < s.

6. With the view of getting a contradiction, assume $\sqrt{p}$ is rational. Then $\sqrt{p}=r/s$ for some integers $s$ and $r$ such that they have no common factor other than one, (that is, $s$ and $r$ are in lowest terms). Squaring both sides and rearranging terms gives $s^2p=r^2.$ Since $p$ is prime, $r$ must be divisible by $p$, say $r=pt$ for some integer $t.$ By substitution, $s^2p=p^2t^2$, so that $s^2=pt^2$, and thus $s$ must also be divisible by $p$, a contradiction.