Prove or Disprove

**harish21** · February 20th 2010, 10:02 AM

The set of squares of rational numbers is inductive.

How do we justify whether this statement is true of false?

**Drexel28** · February 20th 2010, 10:16 AM

Originally Posted by harish21

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?

**harish21** · February 20th 2010, 10:39 AM

Originally Posted by Drexel28

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?

**Drexel28** · February 20th 2010, 10:48 AM

Originally Posted by harish21

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 $\frac{p}{q}+1=\frac{p+q}{q}$ ...

**HallsofIvy** · February 20th 2010, 10:51 AM

Originally Posted by harish21

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?

**Drexel28** · February 20th 2010, 10:52 AM

Originally Posted by Drexel28

Uh...if that is all you need to prove. Note that $1\in\mathbb{Q}$ and $\frac{p}{q}+1=\frac{p+q}{q}$ ...

Originally Posted by HallsofIvy

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!

**HallsofIvy** · February 20th 2010, 10:53 AM

No longer needed

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