Construction of Q/Equivalence classes/Exercise

We consider on ${}\mathbb {Z} \times (\mathbb {Z} \setminus \{0\})$ the relation

$(a,b)\sim (c,d),\,{\text{ if }}ad=bc{\text{ ist}}.$

a) Show that ${}\sim$ is an equivalence relation.

b) Show that for every ${}(a,b)$ there exists an equivalent pair ${}(a',b')$ with ${}b'>0$.

c) Let ${}M$ be the set of all equivalence classes of this equivalence relation. We define a mapping

$\varphi \colon \mathbb {Z} \longrightarrow M,z\longmapsto [(z,1)].$

Show that ${}\varphi$ is injective.

d) Define on ${}M$ (from part c) an operation ${}+$ such that ${}M$, together with this operation and with ${}[(0,1)]$ as neutral element, is a group, and such that for the mapping ${}\varphi$ the relation

${}\varphi (z_{1}+z_{2})=\varphi (z_{1})+\varphi (z_{2})\,$

holds for all ${}z_{1},z_{2}\in \mathbb {Z}$.

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