Prelim Problem # 19 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 $(a,0) <_{\mathbb{Z}} (b,0) \longleftrightarrow a+0 <_{\mathbb{N}} b+0$
$\Longrightarrow (a,0) <_{\mathbb{Z}} (b,0) \longleftrightarrow a <_{\mathbb{N}} b$
$(a,b)\sim (\tilde{a}, \tilde{b})\longrightarrow a+\tilde{b} = b+ \tilde{a}$
$(c,d)\sim (\tilde{c}, \tilde{d})\longrightarrow c+\tilde{d} = d+ \tilde{c}$

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

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