I've got $R=\{a/b \in\Bbb Q\mid b\, \text{ is odd }\}$ and I have to prove that $R$ is a subgroup of $(\Bbb Q,+)$

Question

I've got $R=\{a/b \in\Bbb Q\mid b\, {\rm is}\, {\rm odd }\}$ and I have to prove that $R$ is a subgroup of $(\Bbb Q,+)$

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I've taken two element from $R$, which are: $a=c/d$ and $b=e/f$ (with $d$ and $f$ odd) and I see that $a+(-b)=(cf-ed)/(df)$ is into $R$ owing to the fact that it is in $\Bbb Q$ and $df$ is odd. But to prove that $R$ is a subgroup of $(\Bbb Q,+)$ I also need to check that $R$ is not empty. How can I prove that? Is it trivial to say that $1$ is into $R$, for example?

Answer 1

Yes, to prove something is "NOT TRUE", it is usually sufficient to give one counter-example. As you said $1\ \epsilon\ R$ is one such example which is sufficient.

In logic:

Let there be a statement $P:= R$ is empty.

Then it's negation $\lnot P = R$ is not empty.

So, all you have to do is find a counter-example to disprove $P$, that will simultaneously prove $\lnot P$.

So, yes, it is enough to state that $1\ \epsilon\ R$.

Answer 2

Try proving that the neutral element is in R .That would be the best approach to show that a group is not empty.

Answer 3

Hint: The number $1$ is odd.${}$