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    MHF Contributor harish21's Avatar
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    Prove or Disprove

    The set of squares of rational numbers is inductive.

    How do we justify whether this statement is true of false?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by harish21 View Post
    The set of squares of rational numbers is inductive.

    How do we justify whether this statement is true of false?
    Inductive set in what way? I define inductive set as being a partially ordered set that fulfills the criteria to apply Zorn's lemma. How do you define it?
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  3. #3
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Drexel28 View Post
    Inductive set in what way? I define inductive set as being a partially ordered set that fulfills the criteria to apply Zorn's lemma. How do you define it?
    The book has given no definition. The only thing it states about a set being inductive is that (i) the number 1 should be in S and (ii) if x is in S, then x+1 is also in S. There was a similar question that had asked to prove or disprove that the set of irrational numbers is inductive. That statement was false because 1 is not an irrational number, and this would violate the rule for the set to be inductive.
    I think the question I have posted is also to be done in a similar way, but I cannot think how we are supposed to do it! any idea?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by harish21 View Post
    The book has given no definition. The only thing it states about a set being inductive is that (i) the number 1 should be in S and (ii) if x is in S, then x+1 is also in S. There was a similar question that had asked to prove or disprove that the set of irrational numbers is inductive. That statement was false because 1 is not an irrational number, and this would violate the rule for the set to be inductive.
    I think the question I have posted is also to be done in a similar way, but I cannot think how we are supposed to do it! any idea?
    Uh...if that is all you need to prove. Note that 1\in\mathbb{Q} and \frac{p}{q}+1=\frac{p+q}{q}...
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    Quote Originally Posted by harish21 View Post
    The book has given no definition. The only thing it states about a set being inductive is that (i) the number 1 should be in S and (ii) if x is in S, then x+1 is also in S.
    That is a definition!

    There was a similar question that had asked to prove or disprove that the set of irrational numbers is inductive. That statement was false because 1 is not an irrational number, and this would violate the rule for the set to be inductive.
    I think the question I have posted is also to be done in a similar way, but I cannot think how we are supposed to do it! any idea?
    Is 1 a square number? If not you are done. If it is, then look at 1+ 1= 2. Is 2 a square number?
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Drexel28 View Post
    Uh...if that is all you need to prove. Note that 1\in\mathbb{Q} and \frac{p}{q}+1=\frac{p+q}{q}...
    Quote Originally Posted by HallsofIvy View Post
    That is a definition!


    Is 1 a square number? If not you are done. If it is, then look at 1+ 1= 2. Is 2 a square number?
    Misread the question!
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