**19. (Preliminary for Th, March 15; finalized Sat, March 24) a. Using our definition of the integers Z as a quotient of N×N, propose an appropriate definition of <Z (that is, [(m,n)]<Z[(p,q)] if and only m,n,p,q satisfy ??).**

a. $[(m,n)] <_{\mathbb{Z}} [(p,q)]\longleftrightarrow m,n,p,q$ satisfy $m+q<_{\mathbb{N}} n+p$ is the appropriate definition of $<_{\mathbb{Z}}$

**b. Prove that your proposal is well defined and extends the linear order <N on the natural numbers (i.e., if two integers also happen to be natural numbers, they satisfy <Z if and only if they satisfy <N).**

If $a,b \in \mathbb{N}$, their respective corresponding components in $\mathbb{Z}$ is $(a,0)$ and $(b,0)$.

If $(a,0) <_{\mathbb{Z}} (b,0) \longleftrightarrow a+0 <_{\mathbb{N}} b+0$

$\Longrightarrow (a,0) <_{\mathbb{Z}} (b,0) \longleftrightarrow a <_{\mathbb{N}} b$

$(a,b)\sim (\tilde{a}, \tilde{b})\longrightarrow a+\tilde{b} = b+ \tilde{a}$

$(c,d)\sim (\tilde{c}, \tilde{d})\longrightarrow c+\tilde{d} = d+ \tilde{c}$

$(a,b)<(c,d)\longrightarrow a+d<b+c$

$(b+\tilde{a} - \tilde{b}) + (c+\tilde{d} - \tilde{c}) < (a+\tilde{b} - \tilde{a}) + (d+\tilde{c} - d)$

$2\tilde{a} + 2\tilde{d} +b+c<2\tilde{b} +2\tilde{c} +a+d$

$a+d<b+c$

$2(\tilde{a} +\tilde{d})< 2(\tilde{b} + \tilde{c})$

$\tilde{a} + \tilde{d} < \tilde{b} + \tilde{c}$

$(\tilde{a},\tilde{b})<(\tilde{c},\tilde{d})$

(Prof. Simmons, 4-2-18: The logic is correct, but more wording and fewer arrows would improve the flow.)

**20. (Preliminary for T, March 20; finalized Sat, March 24) Prove that the polynomial ring Z[x] over the integers is countably infinite.**

Well the this ring has one to one correspondence to the integers which has one to one correspondence to the natural numbers. Thus is countably infinite

(Prof. Simmons, 4-2-18: It's not obvious that the set of all polynomials with integer coefficients is in one-to-one correspondence with the integers; the assertion must be proven.)

**22. (Preliminary for T, March 20; finalized Sat, March 24) Let (R,0,1,+,⋅,≤) be an ordered ring with identity.**

**a. Prove that if x>0, then −x<0.
b. Prove that 1>0.**

**Proof of a:**

$x+(-x)>0+(-x)$

$0> -x$

**Proof of b:**

Suppose $x>0$

$\Longrightarrow x(1/x) > 0(1/x)$

$\Longrightarrow 1>0$

(Prof Simmons 4-23-18: x might not have an inverse….)