A question about dense subsets of the real line

[AxiomOfChoice]

Consider the closed interval A = [a,b]\subset \mathbb{R}. Are the only dense subsets of A the set of all rational numbers in A and the set of all irrational numbers in A? Something tells me that there's got to be more than that, but I can't think of any examples.

Thanks in advance for your help.

 

[HallsofIvy]

Well, obviously, the set of, say, rational numbers with some irrational numbers included would be dense, the set of all irrational numbers with some rational numbers included would be dense. Or you could partition the real numbers into any number of disjoint subsets, take rational numbers in some of the subsets and irrational numbers in the others.

 

[maze]

You could basically throw out a bunch of rational or irrational numbers, so long as you don't throw out too many of them. For example, Q\{1} is still dense. You could probably even get away with throwing out countably infinite rational numbers from Q and be OK - throw out "every other" rational number (they are countable so you can order them).

You could also construct other countable dense subsets by adding an irrational number to all rational numbers. eg, {q+pi: q rational}

That is an interesting question if there are other countable dense subsets that are fundamentally different from Q in a meaningful way. I don't know the answer to that. Probably yes?

 

[lurflurf]

That is an interesting question if there are other countable dense subsets that are fundamentally different from Q in a meaningful way. I don't know the answer to that. Probably yes?

What does that mean?
would
Z+sqrt(2)Z={a+sqrt(2)b |a,b integers}
Z[1/2]={a/2^b|a,b integers}
Sin(Z)={sin(a)|a an integer}
qualify?

 

[maze]

What does that mean?
would
Z+sqrt(2)Z={a+sqrt(2)b |a,b integers}
Z[1/2]={a/2^b|a,b integers}
Sin(Z)={sin(a)|a an integer}
qualify?

I don't understand? Those aren't dense. Do you mean rationals Q instead of integers Z?

 

[lurflurf]

I don't understand? Those aren't dense. Do you mean rationals Q instead of integers Z?

What definition of dense are you using? Try something like if A is dense in B, for any b in B and epsilon in (0,infinity), there exist a in A such that d(a,b)<epsilon. That is A contains points arbitrarily close to any point in B.

Example sin(Z) is dense in [-1,1]
since sine is a continuous function with period 2pi our result follows from the fact that
Z+2piZ is dense in R
which follows from the fact that
aZ+bZ is dense in R when a is rational (and not zero) and b is irrational

 

[HallsofIvy]

You could basically throw out a bunch of rational or irrational numbers, so long as you don't throw out too many of them. For example, Q\{1} is still dense. You could probably even get away with throwing out countably infinite rational numbers from Q and be OK - throw out "every other" rational number (they are countable so you can order them).

You could also construct other countable dense subsets by adding an irrational number to all rational numbers. eg, {q+pi: q rational}

That is an interesting question if there are other countable dense subsets that are fundamentally different from Q in a meaningful way. I don't know the answer to that. Probably yes?

AxiomofChoice said in his first post that the set of irrationals is dense in the reals. That is about as "fundamentally different from Q in a meaningful way" as you can get.

You seem to be asking for sets of real numbers that do no involve rational or irrational numbers. There are no such sets- rational and irrational numbers are all we have!

 

[slider142]

You seem to be asking for sets of real numbers that do no involve rational or irrational numbers. There are no such sets- rational and irrational numbers are all we have!

Hmm, how about changing the sieve. What about algebraic vs. transcendental numbers. Are transcendental numbers dense on the real line?
Or going the other way, are undefinable numbers dense in the reals? By Cantor's slash, they're a "pretty large" subset.

 

[maze]

Well I mean I was looking for countable dense subsets, not any dense subset. But lurf's example of Sin(Z) repeated each interval seems like a pretty good example to me.