A question about dense subsets of the real line

In summary: Hmm, how about changing the sieve. What about algebraic vs. transcendental numbers. Are transcendental numbers dense on the real line? :tongue2:Or going the other way, are undefinable numbers dense in the reals? By Cantor's slash, they're a "pretty large" subset. :confused:
  • #1
AxiomOfChoice
533
1
Consider the closed interval [tex]A = [a,b]\subset \mathbb{R}[/tex]. Are the only dense subsets of [tex]A[/tex] the set of all rational numbers in [tex]A[/tex] and the set of all irrational numbers in [tex]A[/tex]? 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.
 
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  • #2
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.
 
  • #3
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?
 
  • #4
maze said:
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?
 
  • #5
lurflurf said:
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?
 
  • #6
maze said:
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
 
  • #7
maze said:
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!
 
  • #8
HallsofIvy said:
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? :-p
Or going the other way, are undefinable numbers dense in the reals? By Cantor's slash, they're a "pretty large" subset. :confused:
 
  • #9
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.
 

Related to A question about dense subsets of the real line

1. What is a dense subset of the real line?

A dense subset of the real line is a set of real numbers that is contained within the real line and has the property that every interval on the real line contains at least one element of the subset. In other words, the subset is "everywhere" in the real line.

2. How do you determine if a set is dense in the real line?

To determine if a set is dense in the real line, you can use the definition mentioned in the previous question. If every interval on the real line contains at least one element of the set, then the set is considered dense in the real line.

3. What is the importance of dense subsets in mathematics?

Dense subsets are important in mathematics because they can be used to approximate any real number. This property is particularly useful in analysis and topology, where dense subsets are used to define concepts such as continuity and completeness.

4. Can a set be dense in the real line but not be a subset of it?

Yes, a set can be dense in the real line without being a subset of it. For example, the set of rational numbers is dense in the real line but is not a subset of the real line since it also includes irrational numbers.

5. Are there any practical applications of dense subsets?

Yes, dense subsets have practical applications in various fields such as engineering, physics, and computer science. In engineering, dense subsets are used to model continuous systems, while in physics, they are used in the study of quantum mechanics. In computer science, dense subsets are used in algorithms for data compression and optimization.

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