Find the cardinality of these sets

Question

Question from my homework im struggling with

Find the cardinality of these sets:

1) the set of all sequences of natural numbers

2) the set of all arithmetic series (difference between 2 numbers is the same,example 11,9,7 ...)

3) the set of all rising arithmetic series (difference between 2 numbers is positive, example 11 13 15...)

My answers:

1) there are $2^{\aleph_0}$ sequences so the answer is $c$

2) what determines a series is the first number, the difference between 2 numbers, and the last number, so you have 3 criteria, $\aleph_0$ options for each, overall - $3\cdot\aleph_0 = \aleph_0$

3) this is a subset of the answer to question 2), so it is also $\aleph_0$.

But I am wrong.

I know I am wrong because the next question is "Show that there is an isomorphism between the answer to question 1 and the answer to question 3".

Please help :)

Answer 1

Your answers look to be correct.

The only way that I see to salvage the correctness of the follow-up question is to take the "set of all sequences of natural numbers" to be talking about finite sequences, i.e. the set:

$$\Bbb N^{<\omega} = \bigcup_{n\in\Bbb N} \Bbb N^n$$

If you are worried about assuming this, you can use a formulation like (outline, add details as needed):

In view of the next question, 1) seems to be talking about the set of all finite sequences. The cardinality of the set of all infinite sequences is $\aleph_0^{\aleph_0} = 2^{\aleph_0}=c$; the cardinality of the set of all finite sequences is $\sum\limits_{n \in \Bbb N} \aleph_0^n = \aleph_0$.

Such a formulation wards you against making wrong assumptions, and makes it clear to whoever grades the homework that the formulation is not as clear as it should be.

I also recommend checking if somewhere in your notes/book(s) "sequence" is defined to be "finite sequence". Having this clear may help avoid confusion in the future.