A problem about elements in a set

[mathlover28]

I just come across difficulties when handling this problem.

I know a must be odd and b must be even. But how would I count how many elements in the set A and B, other than listing out all the elements?

Appreciate your help. Thanks.

 

[pka]

I just come across difficulties when handling this problem.
I know a must be odd and b must be even. But how would I count how many elements in the set A and B, other than listing out all the elements?

In the textbook from which this question comes does \(\mathbb{N}\) contain \(0~?\) or not?

 

[mathlover28]

In the textbook from which this question comes does \(\mathbb{N}\) contain \(0~?\) or not?

In my course, natural number does not contain 0.

 

[pka]

In my course, natural number does not contain 0.

That makes a difference. It means that set \(\mathcal{A}\) contains all odd positive integers \(3,5,7.\cdots 4039\).
How many are in set \(\mathcal{A}~?\)
If you follow the posting guidelines then you must post your work.

 

[mathlover28]

That makes a difference. It means that set \(\mathcal{A}\) contains all odd positive integers \(3,5,7.\cdots 4039\).
How many are in set \(\mathcal{A}~?\)
If you follow the posting guidelines then you must post your work.

There are 2019 elements in set A.

This is what I literally wrote down for this problem:

Note that A contains odd integers {3,5,7,9,...,4039}. For the set A and B, the elements must also be odd.
i.e. Both a and b+1 are odd. Hence b is even and b >= 2.

Then I am struggling do I really need to count all the elements or not.

Any more hints? Thanks.

 

[pka]

There are 2019 elements in set A. This is what I literally wrote down for this problem:
Note that A contains odd integers {3,5,7,9,...,4039}. For the set A and B, the elements must also be odd.
i.e. Both a and b+1 are odd. Hence b is even and b >= 2.
Then I am struggling do I really need to count all the elements or not.

\(15^3=3375\) BUT \(16^3=4096\) this limits the size of numbers in (\\mathcal{A}\cap\mathcal{B}\)
Since \(\mathcal{B}=\{k(2j+1)^3:~\{k,j\}\subset\mathbb{N}\), we can see for example \(k=1~\&~j\le 15\) would give numbers in both sets.
\(k=3~\&~j\le 11\) would give numbers in both sets BUT not \(k=3~\&~j= 13\) would give numbers in both sets.
Can you finish?

 

[pka]

\(15^3=3375\) BUT \(16^3=4096\) this limits the size of numbers in \(\mathcal{A}\cap\mathcal{B}\)
Since \(\mathcal{B}=\{k(2j+1)^3:~\{k,j\}\subset\mathbb{N}\), we can see for example \(k=1~\&~2j+1\le 15\) would give numbers in both sets.
\(k=3~\&~2j+1\le 11\) would give numbers in both sets BUT not \(k=3~\&~2j+1= 13\) would give numbers in both sets.
Can you finish?

EDITED

 

[mathlover28]

EDITED

I almost arrive the final answer. Thanks a lot!