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\textbf{MATH 3100 Project on Constructing the Rationals.}

\textbf{Part 2.}
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\noindent
Due date:11:59 pm Monday April 8.

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\noindent Exercise 2.

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i.)  [Regarding the addition.]  Let the operation $\oplus$ be defined on the equivalence classes by
\begin{eqnarray*}
[(a,b)] \oplus [(c,d)] & = & [(ad + bc, bd)].
\end{eqnarray*}
Show that $\mathbb{Q}$ with the operation $\oplus$ is an abelian group.

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ii.)[Regarding the multiplication.] Define the operation $\otimes$ on the equivalence classes by
\begin{eqnarray*}
[(a,b)] \otimes [(c,d)] & = & [(ac, bd)].
\end{eqnarray*}
Show that $\mathbb{Q}- \{\mbox{the additive identity of } \oplus\}$ (i.e. the set $\mathbb{Q}$ with the additive identity removed) is an abelian group with the operation $\otimes$.

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iii.) [Identifying the integers inside the rationals.] Define $\varphi: \mathbb{Z} \rightarrow \mathbb{Q}$ by $\varphi(z) = [(z,1)]$.  Show that

\qquad \qquad a.) $\varphi$ is 1-to-1.

\qquad \qquad b.) $\varphi$ is a homomorphism from $(\mathbb{Z}, +)$ to $(\mathbb{Q}, \oplus)$.

\qquad \qquad c.) $\varphi$ is a homomorphism from $(\mathbb{Z}, \cdot)$ to $(\mathbb{Q}, \otimes)$.

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