Infinite Operations with Sets

In summary, for the double containment argument, you need to prove that if x is not equal to infinity then there exists a number such that x=1. For the union, you need to prove that it is all reals, and for the intersection, you only need to prove that it is {2i,4i,6i,8i,10i...}.
  • #1
Hi, I am having problems with proving the double containment argument for the following problems. I would appreciate any feedback. Thanks.

Compute [tex]\bigcup[/tex] (i [tex]\in[/tex] I) Ui and [tex]\bigcap[/tex] (i [tex]\in[/tex] I) Ui if

a. I=N, the set of all positive natural numbers and U_n = (-n,n) [tex]\subseteq[/tex] R (U_n is the open interval between -n and n)
i.
[tex]\bigcap[/tex] (i [tex]\in[/tex] I) Ui = (infinity)

How do I prove the double containment argument?
Do I say that if x is not equal to infinity then there exists a number such that x=1?
ii.
[tex]\bigcup[/tex] (i [tex]\in[/tex] I) Ui = (-1,1)

Do I write is x is a member of (-1,1) then there exists a number such that |x| < 1 or 1< 1/|x|

b. I=N, the set of all real numbers and U_r = (0, 1/(1+r^2)) [tex]\subseteq[/tex] R (U_r is the open interval between 0 and 1/(1+r^2))
i.
[tex]\bigcap[/tex] (i [tex]\in[/tex] I) Ui = (0,0)

For the double containment argument do I write what if x is not equal to 0?
ii.
[tex]\bigcup[/tex] (i [tex]\in[/tex] I) Ui = (0, 0.5)
 
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  • #2
in a) the correct answers are that the union is all of R, and the intersection is (-1,1).

The union part is deep and requires the archimedean axiom, that every real number lies in some interval of form (-n,n).

the intersection part is trivial.

in b the answers are again both wrong, the correct answers being the union is (0,1) and the intersection is empty, unless you meant to say the index set was again all positive integers, instead of all reals, in which case the union is indeed (0,1/2).
 
  • #3
mathwonk said:
in a) the correct answers are that the union is all of R, and the intersection is (-1,1).

The union part is deep and requires the archimedean axiom, that every real number lies in some interval of form (-n,n).

the intersection part is trivial.
For the union of (-n,n)

Could i write x [tex]\in[/tex] (-n,n) for some n [tex]\in[/tex] N
x [tex]\in[/tex] (-n,n) [tex]\subseteq[/tex] for all n.
x [tex]\in[/tex] (-n,n) then there is an integer N such that |x|<1
1<1/|x| then (-N,N), x [tex]\in[/tex] N.

For the intersection, do I still have to do soem proof to show that it is all reals?

in b the answers are again both wrong, the correct answers being the union is (0,1) and the intersection is empty, unless you meant to say the index set was again all positive integers, instead of all reals, in which case the union is indeed (0,1/2).

so if it's the set of all reals then the set is empty?

Second question:
a. I={2,4,6,8..} the set of all even numbers and i Ui={ni: nx [tex]\in[/tex] Z}
For the union, is it all reals?
For the intersection is it just {2i,4i,6i...}

b. I={1,3,5,7,9..} the set of all odd numbers and i Ui={ni: nx [tex]\in[/tex] Z}
For the union, is it all reals?
For the intersection is it just {i,3i,5i,7i...}

Thank you
 
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1. What are infinite operations with sets?

Infinite operations with sets refer to mathematical operations that involve an infinite number of elements or members in a set. These operations are often used to manipulate or compare infinite sets, such as the set of all natural numbers or the set of all real numbers.

2. What are some examples of infinite operations with sets?

Some examples of infinite operations with sets include union, intersection, and complement. Union involves combining all elements from two or more sets, intersection involves finding the common elements between two or more sets, and complement involves finding all elements in one set that are not in another set.

3. How are infinite operations with sets different from finite operations?

Infinite operations with sets are different from finite operations in that they involve an infinite number of elements, whereas finite operations involve a finite number of elements. This means that infinite operations can produce different or more complex results, and may require different mathematical techniques to solve.

4. What are some real-world applications of infinite operations with sets?

Infinite operations with sets have various real-world applications, such as in computer science and data analysis. For example, union and intersection can be used to combine or compare large data sets, while complement can be used to identify missing data or outliers. These operations are also used in fields such as economics, physics, and statistics.

5. Are there any limitations to performing infinite operations with sets?

Yes, there are some limitations to performing infinite operations with sets. One limitation is that it is not always possible to perform these operations on infinite sets, as some sets may be too complex or undefined. Additionally, infinite operations may also be computationally intensive and time-consuming, making them difficult to perform in certain scenarios.

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