Infinite subsets within finite sets

[Unity+]

Let's say you have a set A which consists of a subset B. Now, let's say that subset B contains infinite elements within it, but within the bounds that it would be finite in order to fit within the constraints of the finite of A. Would this be possible?

 

I am trying to get me head around the idea that something can be infinitely small, like how there are infinitely many branches within a finite area of a fractal and yet it can't be infinitely larger than the set that contains it because it would seem a paradox would arise. I might need to clarify, so if it is confusing just ask me to clarify this.

[mathematic]

As long as your definitions for elements are consistent, every element of B must be an element of A, so if B has an infinite number, then A must also.

[Unity+]

As long as your definitions for elements are consistent, every element of B must be an element of A, so if B has an infinite number, then A must also.

But, then that must mean a fractal that only takes up a finite area of space is infinitely large because it has infinitely repeating fractal "tails", and therefore by definition has infinite size, but this is not the case.

 

The analogy must not work well though so correct me if I am wrong.

 

EDIT: It can also be argued with the idea that the width of an array of elements could be infinitely large, but its height is finite. The same type of concept can be applied to sets in the case that there is a bound within the subset that may contain infinite elements in one "direction", but in the other it contains finite elements. Therefore, this "direction" determines the finite or infinite property to the subset, which then defines this property to the set that consists of this subset.

Edited January 24, 2014 by Unity+

[studiot]

 

Let's say you have a set A which consists of a subset B. Now, let's say that subset B contains infinite elements within it, but within the bounds that it would be finite in order to fit within the constraints of the finite of A. Would this be possible?

 

You need to distinguish between set enumeration (the number of members of a set) which may be infinite and the extremal values taken on by those members ( the bounds of the set) which will be finite if bounds exist.

 

Does this help?

 

Edit Some good models can be drawn from the set of all real numbers.

 

This is unbounded and infinite (ie contains an infinite number of members)

 

The set of real numbers greater than 1 and less than 2 is a subset of the reals that is still infinite but bounded above (by2 ) and below (by1), although neither 2 nor 1 belong to the subset.

 

The real number 1 is finite and a finite subset of the reals.

 

tan(90) is unbounded, but a finite subset (it contains one member)

Edited January 24, 2014 by studiot

[Unity+]

 

You need to distinguish between set enumeration (the number of members of a set) which may be infinite and the extremal values taken on by those members ( the bounds of the set) which will be finite if bounds exist.

 

Does this help?

It does help in a way for me to clarify my question. I am referring to set enumeration(I think, it might just be related to the values of that member. I might need this clarified first).

 

For example, let's say I have an array list(or matrix) of an infinite size(not "outward", but having a bound)(I just realized that this statement is nonsense).

 

[math]C=\begin{bmatrix} C_{x_{n}}& 0 &\cdots \\ 0& 0 & \cdots \\ \vdots &\vdots &\ddots \end{bmatrix}[/math]

 

Which is contained within another set that is finite.

 

[math]F=\begin{bmatrix} C & 0 &0 \\ 0 & 0 &0 \\ 0 & 0 & 0 \end{bmatrix}[/math]

 

I apologize for the confusing question. The reason why it may be hard to explain could be because it isn't logically possible to do this.

 

EDIT: Would this question be more aimed at the magnitude of the element or set enumeration?(Sorry if am asking so many questions).

Edited January 24, 2014 by Unity+

[studiot]

Look at my edit/ addendum to my previous post.

 

 

(Sorry if am asking so many questions).

 

Don't be sorry, this is a perfectly respectable discussion.

Edited January 24, 2014 by studiot

[Unity+]

Look at my edit/ addendum to my previous post.

Therefore, there is the existence of bounded sets that contain infinite members with the example of all reals that exist between 1 and 2?

 

If this statement is correct, then I understand the idea now. Thanks for the help.

 

EDIT: Would this be the correct notation to denote this kind of set?

 

[math]M\ni \mathbb{R}\rightarrow \left ( 1< x< 2 \right )[/math]

 

EDIT2: I don't try to ask too many questions because it may show some "laziness" in figuring out logic that may be obvious. I usually try to figure this stuff out on my own, but sometimes I hit those barriers of my finite knowledge.

 

I know the common phrase is "No question is a stupid question", but I feel it better to use use known knowledge to figure out more knowledge(though sometimes that doesn't help).

Edited January 24, 2014 by Unity+

[studiot]

Set theory notation is not my speciality, but I am used to the following, which is very common.

 

[math]M = \{ x:1 < x < 2\} [/math]

or

[math]M = \{ x|1 < x < 2\} [/math]

 

Which read in English

 

M is the set of all real numbers, x, which obey the given condition ie greater than 1 but less than 2.

 

I am sure there are other notational variations, but you need the colon or vertical line to separate the variable x and the condition. The list of members or the variable+condition are usually enclosed in parenthesis type brackets.

Edited January 24, 2014 by studiot

[Unity+]

Set theory notation is not my speciality, but I am used to the following, which is very common.

 

[math]M = \{ x:1 < x < 2\} [/math]

 

or

[math]M = \{ x|1 < x < 2\} [/math]

 

 

Which read in English

 

M is the set of all real numbers, x, which obey the given condition ie greater than 1 but less than 2.

 

I am sure there are other notational variations, but you need the colon or vertical line to separate the variable x and the condition. The list of members or the variable+condition are usually enclosed in parenthesis type brackets.

For both notations, wouldn't you have to denote that you want only the real numbers between those two values for x, or is that implied within the notation(especially since whole numbers can't be located within the set anyways given the condition)?

 

Once that question is answered, that will be all for this topic. Thanks for the help, again.

Edited January 24, 2014 by Unity+

[studiot]

Well yes, you would say somewhere else that you were dealing with real numbers.

If you wanted to be strict, or just starting you could add a second condition

[math]M = \{ x:1 < x < 2:x\;real\} [/math]

 

or

[math]M = \{ x:1 < x < 2:x\; \in R\} [/math]

or as many conditions as you needed for that situation.

 

Edit : Sets are usually shown as upper case and members as lower case (as you have done)

 

I will wish you goodnight on that.

Edited January 25, 2014 by studiot

[Someguy1]

Let's say you have a set A which consists of a subset B. Now, let's say that subset B contains infinite elements within it, but within the bounds that it would be finite in order to fit within the constraints of the finite of A. Would this be possible?

 

I am trying to get me head around the idea that something can be infinitely small, like how there are infinitely many branches within a finite area of a fractal and yet it can't be infinitely larger than the set that contains it because it would seem a paradox would arise. I might need to clarify, so if it is confusing just ask me to clarify this.

 

 

It's perfectly ok for an infinite set to be an element of a finite set. For example if R, Q, and Z are the set of real numbers, the set of rational numbers, and the set of integers, respectively, then the set X = {R, Q, Z} is a set containing exactly three elements.

But an infinite set can never be a subset of a finite set, since the cardinality of a subset must be less than or equal to the cardinality of the original set.

Edited January 25, 2014 by Someguy1

[Function]

I don't know if this is what you're looking for, and it might seem too obvious, but I'll just give it a try.

This is an example:

 

[math]A=\left[a,b\right][/math]

[math]B=\left[c,d\right][/math]

There are infinite elements in [math]B[/math] (e.g. [1,2] contains infinite elements: 1,1; 1,01; 1,001; ...; 1,999999999...)

[math]c>a, d<b[/math]

[math]\Rightarrow B\in A[/math]

Edited January 25, 2014 by Function

[studiot]

Function, are you sure this is what you meant?

 

You have said d<c, but offered c = 1, d = 2?

 

 

Someguy1, are you sure of your terminology?

Did you mean proper subset?

 

What are the subsets of X = {R,Q,Z} ?

Edited January 25, 2014 by studiot

[Function]

Function, are you sure this is what you meant?

 

You have said d<c, but offered c = 1, d = 2?

 

sets1.jpg

 

 

Oh no, I meant: [math]c>a[/math] and [math]d<b[/math] and [math]c<d[/math]

[Someguy1]

 

 

Someguy1, are you sure of your terminology?

Did you mean proper subset?

 

What are the subsets of X = {R,Q,Z} ?

I wrote exactly what I meant. What I wrote is mathematically correct. I'm a little puzzled by your comment, since what I wrote is so widely known.

 

There are eight subsets of X, just as there are eight subsets of any set of 3 elements. Do you disagree? The eight subsets are:

 

1. The empty set.

 

2. {R}

 

3. {Q}

 

4. {Z}

 

5. {R, Q}

 

6. {R, Z}

 

7. {Q, Z}

 

8. {R, Q, Z}

Edited January 25, 2014 by Someguy1

[studiot]

 

I wrote exactly what I meant.

 

Thank you for distinguishing more clearly between sets, subsets and members of sets and subsets.

[Unity+]

I'm a little puzzled by your comment, since what I wrote is so widely known.

No need to get all roused up in this discussion. Both of you were right. If he made a mistake or needed clarification, I see no need to make such a comment that is unneeded because even if it is widely known doesn't mean people, like me, know it as well. Making such an assumption is not a very good teaching characteristic. Just my opinion.

Edited January 25, 2014 by Unity+

[Someguy1]

But, then that must mean a fractal that only takes up a finite area of space is infinitely large because it has infinitely repeating fractal "tails", and therefore by definition has infinite size, but this is not the case.

Ah, perhaps do you mean something like taking the unit interval, which has length one, and dividing it into infinitely many nonzero pieces of size 1/2, 1/4, 1/8, 1/16, ... etc? In that way we have infinitely many intervals, each interval has nonzero size, yet their union has size one.

 

Is this what you are asking? I confess I'm having a hard time understanding your specific question.

[Unity+]

Ah, perhaps do you mean something like taking the unit interval, which has length one, and dividing it into infinitely many nonzero pieces of size 1/2, 1/4, 1/8, 1/16, ... etc? In that way we have infinitely many intervals, each interval has nonzero size, yet their union has size one.

 

Is this what you are asking? I confess I'm having a hard time understanding your specific question.

Yes, that is similar to what studiot brought up earlier, where there are infinite real numbers between two real numbers 1 and 2. It is bounded, but infinitely large.

[Someguy1]

Yes, that is similar to what studiot brought up earlier, where there are infinite real numbers between two real numbers 1 and 2. It is bounded, but infinitely large.

No, my example is very different. There are infinitely many real numbers between 1 and 2, but each real number has zero length.

 

In my example, there are infinitely many nonzero lengths whose union is finite.

 

I'm just trying to figure out what you're asking.