What is the cardinality of these sets?

Question

1) A set, that contains finite binary strings. 2) infinite binary strings. 3) finite sequences of natural numbers. 4) infinite sets of natural numbers.

Answer 1

1) If you mean a set contains all finite binary strings then you can map it (bijectively) onto natural number by binary representation and therefore you know the cardinality.

2) Infinite binary strings are equivalent to functions from $\mathbb{N}$ to $\{0,1\}$, so the cardinality (usually denoted by $2^{\mathbb{N}}$) is uncountable. The proof should be similar to http://www.math.cmu.edu/~wgunther/127m12/notes/CSB.pdf

3) It is same as (1). (Consider you choose a set by marking binary numbers, then you permute them in finitely many ways so it will not change the cardinality)

4) It is same as (2).

Answer 2

1) $\aleph_0$

2) $2^{\aleph_0}$

3) $\aleph_0$

4) $2^{\aleph_0}$

The set of all finite binary strings is the union, over all natural numbers $n$, the set of all binary strings of length $n$. Each of those sets is countable (finite even) and the countable union of countable sets is countable.

The set of all infinite binary strings can be naturally bijected with the powerset of the natural numbers and thus has size $2^{\aleph_0}$.

The other two follow similar arguments.