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Old 12-02-2017, 08:18 PM   #1
robert_utk
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Are these two expressions “equal”?

Short and sweet.

Is this an allowed description of a uniform real distribution of numbers?

From a formal mathematics perspective, can I say that:

The random real numbers (0,1)

Equals....

The random real numbers [1-.999999etc,.9999etc]

Does forcing one of the inclusive boundaries to be an expression of the other inclusive boundary break a formal rule?

Thanks!
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Old 12-02-2017, 08:30 PM   #2
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Re: Are these two expressions “equal”?

Well, the problem is, I think, that .9 repeating equals 1. It's not "close to 1", it is 1. So you turned (0, 1) into [0, 1] and those are not the same.

ETA: really I keep going back and forth. I could see an argument either way.
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Old 12-02-2017, 08:42 PM   #3
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Re: Are these two expressions “equal”?

Would both be an acceptable way to say “any number between 0 and 1” ? Granted, the simpler expression should be used for simplicity sake, but can you define one side as an expression of the other, such as (1-1,1)?
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Old 12-02-2017, 08:50 PM   #4
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Re: Are these two expressions “equal”?

Sure? Theoretical math is not real big on doing arithmetic so it's common to see ranges like (1/2, sqrt(2)) and no one is going to make you say (0.5, 1.414...)

But you should prefer to just say (0, 1) imo because this is literally what it's meant for
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Old 12-02-2017, 08:56 PM   #5
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Re: Are these two expressions “equal”?

Is (1-x,x) a valid expression of real numbers, so long as x is real?

Thank you RustyBrooks
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Old 12-02-2017, 09:03 PM   #6
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Re: Are these two expressions “equal”?

Quote:
Originally Posted by robert_utk View Post
Is (1-x,x) a valid expression of real numbers, so long as x is real?
I dunno, seems fine to me. The rules are pretty fluid as long as it's clear what you mean.
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Old 12-02-2017, 09:06 PM   #7
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Re: Are these two expressions “equal”?

Yay!
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Old 12-02-2017, 10:27 PM   #8
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Re: Are these two expressions “equal”?

Pretty sure that in order to say (1-x, x) you have to specify that x >= 1/2 (if x=1/2 then that's the null set so that works).
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Old 12-03-2017, 12:52 PM   #9
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Re: Are these two expressions “equal”?

Quote:
Originally Posted by RustyBrooks View Post
Well, the problem is, I think, that .9 repeating equals 1. It's not "close to 1", it is 1. So you turned (0, 1) into [0, 1] and those are not the same.

ETA: really I keep going back and forth. I could see an argument either way.
There is not a (correct) argument either way. 0.9999 repeating is a real number and 1 is a real number. If 0.99999 repeating is less than 1 there exists a rational number x such that 0.99999 repeating < x < 1. No such rational number exists, so 0.9999 repeating equals 1, by the density of rational numbers.

If you were not saying you go back and forth on if they are equal, I apologize for misreading what you meant.
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Old 12-03-2017, 10:54 PM   #10
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Re: Are these two expressions “equal”?

No, they are definitely equal.

But consider, what's the largest number in the range (0, 1)? If you were going to rewrite the expression (0, 1) as [0, x], what's x? I can't think of a better candidate than .9 repeating.
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Old 12-03-2017, 11:26 PM   #11
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Re: Are these two expressions “equal”?

There is no largest number in the range, so you can't rewrite it in square brackets except as the expression
[0,1] \ [0,0]U[1,1]
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Old 12-04-2017, 03:08 AM   #12
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Re: Are these two expressions “equal”?

Quote:
Originally Posted by RustyBrooks View Post
No, they are definitely equal.

But consider, what's the largest number in the range (0, 1)? If you were going to rewrite the expression (0, 1) as [0, x], what's x? I can't think of a better candidate than .9 repeating.
There is no "largest number" in that set. A set doesn't have to have a maximum. For instance, what's the largest number in R?

The (0,1) range does have an infimum and a supremum.

You can't find an x such as (0,1) and (0,x] are the same set. If you could, there wouldn't any need of discriminate between "(" and "[" in the interval notation.
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Old 12-04-2017, 10:35 AM   #13
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Re: Are these two expressions “equal”?

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Originally Posted by nickthegeek View Post
There is no "largest number" in that set. A set doesn't have to have a maximum. For instance, what's the largest number in R?

The (0,1) range does have an infimum and a supremum.

You can't find an x such as (0,1) and (0,x] are the same set. If you could, there wouldn't any need of discriminate between "(" and "[" in the interval notation.


I knew you guys and gals would know the answer, and help me learn as well! I know I should crack a book and learn such things on my own. However, my time is limited and devoted to a different problem.

What I *should* have asked was:

Can I say that....

(0,1) is interchangeable with (1-1,1)

Which I think is yes, from the help here from RustyBrooks
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Old 12-04-2017, 04:14 PM   #14
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Re: Are these two expressions “equal”?

Quote:
Originally Posted by robert_utk View Post
Can I say that....

(0,1) is interchangeable with (1-1,1)
I can hardly see the point of this, but you can of course, since 1-1=0. (0,1) is even interchangeable with (44-44,73-72), but I can't see how this "result" can be of any use.
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Old 12-06-2017, 02:54 PM   #15
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Re: Are these two expressions “equal”?

When you want to relate the results of a problem involving (0,1) to the theoretical results of the same problem in (0,x) where x>1 then a particular nomenclature will arise. Thus the bounds become an expression of each other that are no as simple as just saying (0,x+1) if the new bounds are in fact (x-1,2x) or somesuch
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