Question about the set of irrationals.

  • Thread starter cragar
  • Start date
  • Tags
    Set
In summary, the conversation discusses the possibility of having a set containing all irrational numbers with measure zero and the measure of irrational numbers and rational numbers. It is mentioned that using the Lebesgue measure, it is not possible to have a set with all irrational numbers and measure zero, while using the zero measure allows for such a set. The concept of infinite measure is also discussed, with it being noted that in measure theory, there is only one type of infinity. The conversation also touches on Cantor cardinalities and the power set having a higher cardinality than the original set.
  • #1
cragar
2,552
3
Is it possible to have a set that contain all the irrationals that has measure zero.
I don't know that much about measure theory. Or I guess we could just ask what is the measure of the irrationals. I know it is possible to have uncountable sets that have measure zero.
 
Physics news on Phys.org
  • #2
cragar said:
Is it possible to have a set that contain all the irrationals that has measure zero.
I don't know that much about measure theory. Or I guess we could just ask what is the measure of the irrationals. I know it is possible to have uncountable sets that have measure zero.

The measure of the set of irrational numbers is one.

So the measure of the rationals must be zero.

I seem to recall that the measure of any countable set is zero.
 
  • #3
You didn't specify which measure you are talking about, so if we use the zero measure, then the answer is yes.

If you are talking about using the Lebesgue measure, the answer is no. Let A be all rationals between 0 and 1, and B be all irrationals between 0, and 1. Then
##m(A) + m(B) = m([0,1]) = 1##
But ##m(A) = 0##, so B has non-zero measure.

Note that if C is any set that contains every irrational between 0 and 1 then ##m(C) \geq m(B)## by monotonicity.
 
  • #4
ImaLooser said:
The measure of the set of irrational numbers is one.

What? That's not true, is it?
 
  • #5
I suspect that Imalooser meant the set of all irrational numbers between 0 and 1.
 
  • #6
HallsofIvy said:
I suspect that Imalooser meant the set of all irrational numbers between 0 and 1.

That would make sense.
 
  • #7
HallsofIvy said:
I suspect that Imalooser meant the set of all irrational numbers between 0 and 1.

Sorry, I'm so used to probability measures that I simply assumed that. Probability measures are normed so that the maximum measure is always 1 and the minimum is zero. Duh.
 
  • #8
do they have sets with different infinite measure?
 
  • #9
cragar said:
do they have sets with different infinite measure?

What do you mean with "they"??

Do you mean whether probability spaces have sets of infinite measure? The answer is no: the largest possible measure is 1.
 
  • #10
I guess I mean in ZFC are their sets that have infinite measure.
But I guess you said they dont
 
  • #11
cragar said:
I guess I mean in ZFC are their sets that have infinite measure.

First you need to specify what you mean with "measure". Which measure are you talking about? If you're talking about Lebesgue measure on [itex]\mathbb{R}[/itex] (which is basically the rigorous version of length), then there are sets of infinite measure. The set [itex]\mathbb{R}[/itex] itself has infinite measure.

But I guess you said they dont

I didn't say that. I said that in a probability space (that is when you work with a probability measure), then all sets have finite measure by definition. But when not working with a probability measure, then there might be sets of infinite measure.
 
  • #12
ok thanks for your response. Are their sets that have larger Lebesgue measure
than the set of reals.
 
  • #13
cragar said:
ok thanks for your response. Are their sets that have larger Lebesgue measure
than the set of reals.

No, since they already have infinite measure.

In measure theory, there is only one kind of infinity. There is not an entire class of infinities like the infinites of Cantor.
 
  • #14
cragar said:
ok thanks for your response. Are their sets that have larger Lebesgue measure
than the set of reals.

Are you thinking of Cantor cardinalities? If so, the answer is yes. You can take the power set of any set, and the power set will have higher cardinality.
 
  • #15
ImaLooser said:
Are you thinking of Cantor cardinalities? If so, the answer is yes. You can take the power set of any set, and the power set will have higher cardinality.

cragar explicitly asked about the Lebesgue measure, not cardinality.

To further what micromass said, the Lebesgue measure is a specific measure defined on certain subsets of ℝ. Therefore there can't be a Lebesgue measurable set that has measure larger than ##m(\mathbb{R})##.
 

1. What is the set of irrationals?

The set of irrationals is a subset of the real numbers that includes all numbers that cannot be expressed as a ratio of two integers. These numbers are considered irrational because they cannot be written as a decimal or fraction with a finite or repeating pattern.

2. How do you identify a number as irrational?

A number can be identified as irrational if it cannot be expressed as a ratio of two integers. This can be determined by attempting to write the number as a decimal or fraction and looking for a repeating or terminating pattern. If no such pattern exists, the number is irrational.

3. What is the difference between irrational and rational numbers?

Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot. Rational numbers have either repeating or terminating decimals, while irrational numbers have non-repeating, non-terminating decimals. Additionally, rational numbers can be written as fractions, while irrational numbers cannot.

4. Are there any patterns or relationships among irrational numbers?

Unlike rational numbers, there are no patterns or relationships among irrational numbers. Each irrational number is unique and cannot be expressed as a simple fraction or ratio. However, there are some special irrational numbers, such as pi and e, that have important and interesting mathematical properties.

5. How are irrational numbers useful in mathematics and science?

Irrational numbers are essential in mathematics and science because they allow us to accurately represent and work with non-terminating and non-repeating quantities. They are used in fields such as geometry, physics, and statistics, and are important for advanced calculations and mathematical proofs.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
14
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
8
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
22
Views
2K
Replies
15
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
  • Classical Physics
3
Replies
85
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
959
  • Set Theory, Logic, Probability, Statistics
2
Replies
55
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
334
  • Set Theory, Logic, Probability, Statistics
Replies
13
Views
934
Back
Top