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

[ver_mathstats]

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.

 

[PeroK]

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?

 

[ver_mathstats]

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 A_j={-j.j}? Thank you for the reply.

 

[LCKurtz]

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.

 

[ver_mathstats]

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.

 

[LCKurtz]

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?