Questions about the union of these sets?

[Mates]

I was told that the union of [0.9], [0.09], [0.009], [0.0009] ... is [0.999 ...]

1) Is this true?

2) If so, then would this be a real number, why or why not?

3) If it is real, then isn't it the next real number less than 1 (which isn't suppose to exist).

4) Finally, what does this say about 0.999 ... = 1. If 1 isn't in the union, then isn't there a contradiction here in that it doesn't equal 1?

 

[Dr.Peterson]

I was told that the union of [0.9], [0.09], [0.009], [0.0009] ... is [0.999 ...]

1) Is this true?

2) If so, then would this be a real number, why or why not?

3) If it is real, then isn't it the next real number less than 1 (which isn't suppose to exist).

4) Finally, what does this say about 0.999 ... = 1. If 1 isn't in the union, then isn't there a contradiction here in that it doesn't equal 1?

How are you defining [0.9] as a set? You must be using some notation I am not familiar with.

What does it mean to take the union of these sets?

Also, who told you this? Can you quote exactly what they said, and what proof they gave?

 

[Mates]

How are you defining [0.9] as a set? You must be using some notation I am not familiar with.

What does it mean to take the union of these sets?

Also, who told you this? Can you quote exactly what they said, and what proof they gave?

https://www.reddit.com/r/math/comments/18hle93/_/kdbjxpb

On reddit, my name is a_vat_in_the_brain and I had a conversation with just one poster (ace||of||spades)

We were actually talking about intervals, and the poster used the term "overlay" instead of union sometimes.

I messed up because I thought it would work for sets.

 

[Dr.Peterson]

We were actually talking about intervals, and the poster used the term "overlay" instead of union sometimes.

I messed up because I thought it would work for sets.

An interval is a set. But you used something other than interval notation. Do you not see the difference?

He talked about intervals like [0, 0.1], which means the set of all x such that [imath]0\le x\le 0.1[/imath]. Your notation [0.9] has only one number; if I took it to mean anything, it would be identical to {0.9}.

I don't see anything there that looks like what you asked, so I still don't know what to say about your questions. Do you understand interval notation, and sequences, and sequences of intervals? Do you know about the concept of limits?

 

[Mates]

An interval is a set. But you used something other than interval notation. Do you not see the difference?

He talked about intervals like [0, 0.1], which means the set of all x such that [imath]0\le x\le 0.1[/imath]. Your notation [0.9] has only one number; if I took it to mean anything, it would be identical to {0.9}.

I don't see anything there that looks like what you asked, so I still don't know what to say about your questions. Do you understand interval notation, and sequences, and sequences of intervals? Do you know about the concept of limits?

Then I want to change the word "set" to the word "interval" in the title of the OP. I was just using one number to try to really focus on it.

Having said that, after we overlay the intervals, aren't we left with the number 0.999 ...?

 

[pka]

I think you want [imath][0,0.9]\cup[0,0.09]\cup[0,0.009][/imath]. Is it not that?
However, that union is just [imath][\bf{0,0.9]}[/imath] because [imath][0,0.009]\subset[0,0.09]\subset[0,0.9][/imath]
Do you follow that?

[imath][/imath][imath][/imath]

 

[Dr.Peterson]

Then I want to change the word "set" to the word "interval" in the title of the OP. I was just using one number to try to really focus on it.

Having said that, after we overlay the intervals, aren't we left with the number 0.999 ...?

Huh? A set is an interval; that's not the issue. [0.9] is not an interval any more than it is any sort of set!

Please restate the whole question, using meaningful notation and terminology.

I think what you mean may be the union of the intervals [0, 0.9], [0, 0.99], [0, 0.999], and so on. This union will include every number less than 1, but not 1 itself. (But it's not at all what you wrote.)

But that is not relevant to what appears to be your ultimate question, whether 0.999..., which is the limit of the numbers 0.9, 0.99, 0.999, ..., is equal to 1. It is.

 

[Mates]

I think you want [imath][0,0.9]\cup[0,0.09]\cup[0,0.009][/imath]. Is it not that?
However, that union is just [imath][\bf{0,0.9]}[/imath] because [imath][0,0.009]\subset[0,0.09]\subset[0,0.9][/imath]
Do you follow that?

[imath][/imath][imath][/imath]

I made a mistake. I should have put [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... is [0, 0.9999 ...]

Doesn't this equal 0.9999 ...?

 

[Mates]

Huh? A set is an interval; that's not the issue. [0.9] is not an interval any more than it is any sort of set!

Please restate the whole question, using meaningful notation and terminology.

I think what you mean may be the union of the intervals [0, 0.9], [0, 0.99], [0, 0.999], and so on. This union will include every number less than 1, but not 1 itself. (But it's not at all what you wrote.)

But that is not relevant to what appears to be your ultimate question, whether 0.999..., which is the limit of the numbers 0.9, 0.99, 0.999, ..., is equal to 1. It is.

Like said in my post to Pka,
I should have put [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... is [0, 0.9999 ...]

Doesn't this equal 0.9999 ...?

 

[Dr.Peterson]

Like said in my post to Pka,
I should have put [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... is [0, 0.9999 ...]

Doesn't this equal 0.9999 ...?

No, an interval is not the same as a number! Why do you think that???

 

[blamocur]

Like said in my post to Pka,
I should have put [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... is [0, 0.9999 ...]

Doesn't this equal 0.9999 ...?

Do you that the union of those interval equals [imath][0, 0.9999...][/imath]? I wouldn't agree with that either because [imath]0.999...[/imath] is equal to 1.0, but the union is equal to [imath][0, 1)[/imath], not [imath][0, 1][/imath].

 

[Mates]

No, an interval is not the same as a number! Why do you think that???

It's not the segment that I want to focus in on, I am just interested in the number 0.999 ... itself. (This is why I tried to use only 0.999 ... in the OP, but that didn't work)

 

[Mates]

Do you that the union of those interval equals [imath][0, 0.9999...][/imath]? I wouldn't agree with that either because [imath]0.999...[/imath] is equal to 1.0, but the union is equal to [imath][0, 1)[/imath], not [imath][0, 1][/imath].

But is taking the union a way to force 0.999 ... to exist on the real number line whether it is equal to 1 or not?

 

[Dr.Peterson]

It's not the segment that I want to focus in on, I am just interested in the number 0.999 ... itself. (This is why I tried to use only 0.999 ... in the OP, but that didn't work)

Then don't talk about segments and unions. Say what you actually mean. What are your thoughts about this?

The truth is that 0.999... is defined as the limit of 0.999...9 (that is, of the sequence {0.9, 0.99, 0.999, 0.9999, ...}, and that limit is equal to 1.

The fact that none of the terms in the sequence is equal to 1 is irrelevant; actually, it's to be expected, as that's how limits work.

Again, the union of those segments is irrelevant; limits are different from unions.

 

[Mates]

Then don't talk about segments and unions. Say what you actually mean. What are your thoughts about this?

The truth is that 0.999... is defined as the limit of 0.999...9 (that is, of the sequence {0.9, 0.99, 0.999, 0.9999, ...}, and that limit is equal to 1.

The fact that none of the terms in the sequence is equal to 1 is irrelevant; actually, it's to be expected, as that's how limits work.

Again, the union of those segments is irrelevant; limits are different from unions.

I don't want to talk about limits because I understand what the limit would say about this. I am interested in exploring what the union produces. Can the union [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... produce the interval [0, 0.999 ...]?

 

[blamocur]

But is taking the union a way to force 0.999 ... to exist on the real number line whether it is equal to 1 or not?

Sorry, but I don't know how to translate this question to formal math. Sounds rather philosophical to me, but philosophy is not my thing.

 

[Dr.Peterson]

I don't want to talk about limits because I understand what the limit would say about this. I am interested in exploring what the union produces. Can the union [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... produce the interval [0, 0.999 ...]?

Then what did you mean when you said

It's not the segment that I want to focus in on, I am just interested in the number 0.999 ... itself.

The number is defined as a limit. It is not defined by a union.

If you really are interested only in the union of those intervals, then you've been told the answer: The union is the interval [0, 0.999...), which is identical to [0, 1). It does not include 0.999... = 1, because that is not in any of the individual intervals.

It appears that you want to assume that an infinite union behaves in a way that it does not. In general, infinity does not tend to behave like finite numbers.

 

[pka]

I don't want to talk about limits because I understand what the limit would say about this. I am interested in exploring what the union produces. Can the union [0, 0.9], [0, 0.99], [0, 0.999], [0, 0.9999] ... produce the interval [0, 0.999 ...]?

We use to spend a great deal of time on this topic in our foundations course.
For example: [imath]\displaystyle\bigcup\limits_{k = 1}^5 {\left[ {0,1 - \frac{1}{{{{10}^k}}}} \right]}=[0,.9]\cup[0,.99]\cup[0,.999]\cup[0,.9999]\cup[0,.99999]=[0,.99999] [/imath]
So that we get [imath]\displaystyle\bigcup\limits_{k = 1}^{\infty} {\left[ {0,1 - \frac{1}{{{{10}^k}}}} \right]}=\left[0,0.\overline{\,9~}\right) [/imath] Note that this is a half-open interval (one not included).

[imath][/imath][imath][/imath][imath][/imath]

 

[Mates]

Then what did you mean when you said

The number is defined as a limit. It is not defined by a union.

If you really are interested only in the union of those intervals, then you've been told the answer: The union is the interval [0, 0.999...), which is identical to [0, 1). It does not include 0.999... = 1, because that is not in any of the individual intervals.

It appears that you want to assume that an infinite union behaves in a way that it does not. In general, infinity does not tend to behave like finite numbers.

Okay, thanks. I just wanted to know if that poster was correct in conversation I had. The poster wrote that the union is in a fully closed interval as [0, 0.999...]. But it seems unanimous here that it is half open as you have.

 

[Mates]

Just out of curiosity, does the union of [0, 0.3], [0, 0.33], [0, 0.333], [0, 0.3333] ... equal [0, 0.333 ...] or put another way as [0, 1/3]?