Examples of Partitions: How to Divide Nonzero Integers into Infinite Sets?

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In summary: Thank you.Are you asking us or telling us? What are the properties of a partition? Do your ##A_1## and ##A_2## satisfy those properties?Thank you.
  • #1
ver_mathstats
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Homework Statement


Show an example of a partition of the nonzero integers into two infinite sets. Show an example of a partition of the nonzero integers into infinitely many sets, such that each set of the partition contains exactly two elements.

2. Homework Equations

The Attempt at a Solution


I know that a partition A is a collection of subsets {Ai}.

An example of a partition of the nonzero integers into two infinite sets would be A1={k∈ℤ: 2k, k≠0} and A2={m∈ℤ:2m+1, m≠0}, so that would mean ℙ={A1, A2}. Am I on the write track with this? I assume since ever number is either odd or even then I could write two sets with even and odd numbers that are infinite but do not include 0.

I am confused how to give an example for the second part however?

Thank you.
 
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  • #2
ver_mathstats said:
I am confused how to give an example for the second part however?

Thank you.

No ideas at all?

What about partitioning the set ##\{-3, -2, -1, 1, 2, 3 \}## into subsets, each with two members? Could you do that?
 
  • #3
PeroK said:
No ideas at all?

What about partitioning the set ##\{-3, -2, -1, 1, 2, 3 \}## into subsets, each with two members? Could you do that?
Oh okay that would make sense, I think the fact it was infinite is what threw me off. So we would just do Aj={-j.j}? Thank you for the reply.
 
  • #4
ver_mathstats said:

Homework Statement


Show an example of a partition of the nonzero integers into two infinite sets. Show an example of a partition of the nonzero integers into infinitely many sets, such that each set of the partition contains exactly two elements.

2. Homework Equations

The Attempt at a Solution


I know that a partition A is a collection of subsets {Ai}.

There is more to it than that though. It isn't just any old collection of subsets.

An example of a partition of the nonzero integers into two infinite sets would be A1={k∈ℤ: 2k, k≠0} and A2={m∈ℤ:2m+1, m≠0}, so that would mean ℙ={A1, A2}. Am I on the write track with this? I assume since ever number is either odd or even then I could write two sets with even and odd numbers that are infinite but do not include 0.
What you have given is not a partition. You don't have ##1## in either subset.
 
  • #5
LCKurtz said:
There is more to it than that though. It isn't just any old collection of subsets.What you have given is not a partition. You don't have ##1## in either subset.
Oh okay, sorry, I still am new to what partitions are. A1={xεZ:x>0} and A2={xεZ:x<0}, for every x∈Z* so P={A1, A2}? Thank you for the reply.
 
  • #6
ver_mathstats said:
Oh okay, sorry, I still am new to what partitions are. A1={xεZ:x>0} and A2={xεZ:x<0}, for every x∈Z* so P={A1, A2}? Thank you for the reply.
Are you asking us or telling us? What are the properties of a partition? Do your ##A_1## and ##A_2## satisfy those properties?
 

Related to Examples of Partitions: How to Divide Nonzero Integers into Infinite Sets?

1. What is a partition in computer science?

A partition in computer science refers to the division of a computer's hard drive into separate sections, each of which can be used to store data and run programs independently. This allows the computer to organize and manage data more efficiently.

2. What are some examples of partitions?

Some common examples of partitions include the C: drive and D: drive on a Windows computer, or the /home and /usr partitions on a Linux system. These partitions are typically used to store the operating system, programs, and personal files separately.

3. How are partitions created?

Partitions can be created using disk partitioning software, such as Disk Management on Windows or Disk Utility on Mac. These tools allow users to allocate space on their hard drive for new partitions and format them for use.

4. Can partitions be resized or deleted?

Yes, partitions can be resized or deleted using disk partitioning software. However, it is important to note that resizing or deleting a partition may result in data loss, so it is important to backup important files before making any changes.

5. Are partitions necessary for a computer to function?

No, partitions are not necessary for a computer to function. However, they can be useful for organizing and managing data, as well as for dual-booting multiple operating systems on the same computer.

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