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hello i have a problem with exercise the problem is follow:

Consider the set $Q$ of integers defined as follows:

$1 ∈ Q$

If $b ∈ Q$, then $2b-1 ∈ Q$

If $b ∈ Q$, then $2b +1 ∈ Q$

What is the total $Q$? Prove your answer carefully.

I have found a solution but I have not shown with any particular methodology. Is there concrete evidence; thank you very much

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    Please show what you have tried and ask a concrete question.2011-11-23
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    Once you have your answer, the proof would be by induction.2011-11-23
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    You can reason as follows: $1 \in Q$ so $2(1)+1=3 \in Q$ and so on. Thus all positive odd numbers are in $Q$. Likewise, $1 \in Q$ so $2(1)-1=1$, $2(3)-1=5$ and so on. Thus we don't get any new numbers from this second condition. $Q$ is the set of positive odd integers. You will need to use [induction](http://en.wikipedia.org/wiki/Mathematical_induction) to formulate a solid proof (avoiding the use of phrases like "and so on").2011-11-23
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    @BillCook: As $1 \in Q$, from the first condition it follows that $3 \in Q$. If we apply the first condition again, we have $7 \in Q$, not $5 \in Q$. We get the $5 \in Q$ by applying the second condition on $3 \in Q$.2011-11-23
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    @Huy Oops! Thanks. I guess I unintentionally demonstrated the danger of using "and so on" :)2011-11-23

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