Examples of partitions

  • #1
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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.
 

Answers and Replies

  • #2
PeroK
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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
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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
LCKurtz
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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}.

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
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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
LCKurtz
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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?
 

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