Prove \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} is Countable

  • Thread starter The Captain
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In summary, the task is to prove that the Cartesian product of three sets of positive integers, written as \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+}, is countable. There are multiple ways to do this, including showing a bijection from N to ZxZ and then another from ZxZ to ZxZxZ. Another approach is to prove that the function f(a,b)=2^{a-1}(2b-1) is a bijection, as it maps ZxZ to ZxZ. This task may be challenging for students who have not taken Abstract Algebra.
  • #1
The Captain
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Homework Statement


Prove that [tex] \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} [/tex] is countable, where X is the Cartesian product.

Homework Equations


The Attempt at a Solution


I'm lost as to where to start proving this.
 
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  • #2
The Captain said:

Homework Statement


Prove that [tex] \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} [/tex] is countable, where X is the Cartesian product.


Homework Equations





The Attempt at a Solution


I'm lost as to where to start proving this.

do you know how to prove this

[tex] \mathbb Z^{+} X \: \mathbb Z^{+} [/tex]

is countable?
 
  • #3
There are many ways to do this you show that there it bijection from N to ZxZ ,that is , ZxZ is countable. Then show that there is bijection from ZxZ to ZxZxZ.

EDIT
someone beat me to it.
 
  • #4
I had to prove that [tex] \mathbb Z^{+} \: X \: Z^{+} \: \rightarrow \: Z^{+} [/tex] was one-one and onto using [tex] f(a,b)=2^{a-1}(2b-1)[/tex], does that count for proving it's countable, and if it's not, no I don't know how to prove it's countable.

The class I'm taking is a giant leap from Calc 4, and Abstract Algebra isn't even pre-req though it probably should be because the professor keeps asking who's taking it before.
 
  • #5
Yes, that is a bijection. So you have already shown the first part all you need to do provide a bijection from ZxZxZ to ZxZ .
 

1. What does it mean for a set to be countable?

A set is said to be countable if its elements can be put into a one-to-one correspondence with the positive integers (1, 2, 3, ...). This means that the set can be enumerated or listed in a specific order.

2. How do you prove that a set is countable?

To prove that a set is countable, we must show that its elements can be put into a one-to-one correspondence with the positive integers. This can be done by explicitly listing the elements in a specific order or by showing that the set can be mapped onto the positive integers.

3. What is meant by the notation \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+}?

The notation \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} represents the Cartesian product of three sets, where \mathbb Z^{+} represents the set of positive integers. This means that the elements of the new set are ordered triplets, where the first element comes from the first set, the second element comes from the second set, and the third element comes from the third set.

4. Why is \mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} considered countable?

\mathbb Z^{+} X \: \mathbb Z^{+} X \: \mathbb Z^{+} is considered countable because its elements can be put into a one-to-one correspondence with the positive integers. This can be done by listing the elements in a specific order, such as (1, 1, 1), (1, 1, 2), (1, 1, 3), ... (1, 2, 1), (1, 2, 2), (1, 2, 3), ... and so on.

5. Can you give an example of a set that is not countable?

Yes, the set of real numbers (represented by \mathbb R) is an example of a set that is not countable. This is because it is impossible to list or enumerate all the real numbers in a specific order, as there are an infinite number of them and they are not discrete like the positive integers.

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