Finalized 19,20, 22 J.O

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….)

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