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\begin{center} \textbf{Project on constructing the rational numbers $\mathbb{Q}$.} \end{center}

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Define the set $\mathbb{Q}$ of pairs of integers as follows: $\mathbb{Q} = \mathbb{Z} \times ( \mathbb{Z} - \{0\})$.  Equivalently $(x,y) \in \mathbb{Q}$ if and only if $x, y \in \mathbb{Z}$ and $y \ne0$.  We define the following relation $\sim$ on $\mathbb{Q}$
\begin{eqnarray*}
(a,b) \sim (c,d) & \mbox{if and only if} & ad = bc.
\end{eqnarray*}

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\begin{center} \textbf{Project on the rational numbers II.} \end{center}

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Part II.  Due by midnight Monday April 6.

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Define $\mathbb{Q}_\sim = \{ [(a,b)] | (a,b) \in \mathbb{Q}\}$.

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\noindent
Problem 3a. Show that there is an ``additive'' identity in $\mathbb{Q}_\sim $ for the operation $+_\sim$.

In other words, show that there exists $(e_1, e_2) \in \mathbb{Q}$ so that for an arbitrary $(a,b) \in \mathbb{Q}$ we have
\begin{eqnarray*}
[(e_1,e_2)] +_\sim [(a,b)] & = & [(a,b)].
\end{eqnarray*}

Note that since (it is easy to see that) the operation is commutative, it's not necessary so show that the identity that we select is both a left identity and a right identity (and similarly below for inverses).

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\noindent
Problem 3b. Show that there is an additive inverse for each element of $\mathbb{Q}$.

In other words, show that for an arbitrary $(a,b) \in \mathbb{Q}$, there exists $(c, d) \in \mathbb{Q}$ so that we have
\begin{eqnarray*}
[(c,d)] +_\sim [(a,b)] & = & [(e_1,e_2)].
\end{eqnarray*}

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\noindent
Problem 4a. Show that there is a ``multiplicative'' identity in $\mathbb{Q}_\sim$ for the operation $\cdot_\sim$.

In other words, show that there exists $(\ell_1, \ell_2) \in \mathbb{Q}$ so that if $(a,b) \in \mathbb{Q}$,  we have
\begin{eqnarray*}
[(\ell_1, \ell_2)] \cdot_\sim [(a,b)] & = & [(a,b)].
\end{eqnarray*}

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\noindent
Problem 4b. Show that there is a multiplicative inverse for each element of $\mathbb{Q}_\sim$ except for the additive identity.


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\noindent
Problem 4c. Show that the additive identity does not have a multiplicative inverse.

I.e.: show that $[(e_1, e_2)]$ does not have a multiplicative inverse.

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\noindent
Problem 5. Show that there is an isomorphic copy of the integers in $\mathbb{Q}_\sim$ which is an isomorphism with respect to both addition ($+_\sim$) and multiplication ($\cdot_\sim$) operations.

For future needs, lets call the isomorphism $\theta$.

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Note that once we's done our proofs for problem 5, then we may stop subscripting the relationships $+_\sim$ and $\cdot_\sim$ since, where appropriate, they are isomorphic to the operations on the integers.

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We need to show that the classes from the equivalence relation $\sim$ on $\mathbb{Q}$ satisfy the same axioms as the integers with regard to the operations $+$ and $\cdot$ (note that I've omitted the subscript) of the operations on the equivalence classes.  We will prove that a selected few ``axioms'' follow from our equivalence relation.  

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\noindent
Problem 6. Show that the operation $+$ is associative.  [Note that by $+$, I mean $+_\sim$ and similarly with $\cdot$.]

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\noindent
Problem 7. Show that the distribution axiom holds:
\begin{eqnarray*}
[(x,y)]\cdot ([(a,b)]) + [(c,d)] ) & = & [(x,y)]\cdot [(a,b)] + [(x,y)]\cdot [(c,d)].
\end{eqnarray*}

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\noindent
Problem 8. Define the inequality $<_\sim$ on the equivalence classes of $\mathbb{Q}$ (i.e. on $\mathbb{Q}_\sim$) so that the following hold:

a.  Define $<_\sim$ so that if $\theta: \mathbb{Z} \rightarrow \mathbb{Q}_\sim$ is the isomorphism defined in part 2 of the project, then $x < y$ if and only if $\theta(x) <_\sim \theta(y)$.

b.  Prove that $<_\sim$ is well-defined.

c. Prove that the relation $<_\sim$ satisfies axioms D4 and D5.

\noindent [Hint: show that for each $(a,b) \in \mathbb{Q} $ there exist $(a',b') \in \mathbb{Q} $ so that $b' >0$ and  $(a,b)   \sim  (a', b')$.  And use the big hint of part I.]









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