Is every element of the power set of an infinite set (e.g. the natural numbers) a finite set?

Question

Is every element of the power set of an infinite set (e.g. the natural numbers) a finite set? Is there a bijection between every element of that power set with the same cardinality?

I am trying to make claims about the existence of a bijection between two subsets of the natural numbers (i.e. the existence of a bijection between two elements of the power set of the natural numbers).

I know that if X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements.

But what about for infinite sets (or, in particular, countably infinite sets)? For example, how would I be able to show that there exists a bijection between every element of the power set of the natural numbers and itself? Just by my own intuition, this seems to be obviously true, but I cannot directly apply the above theorem unless I can say that every subset of the natural numbers is finite (which does not even seem to be true, but I'm not sure).

Any suggestions?

Answer 1

Is every element of the power set of an infinite set (e.g. the natural numbers) a finite set?

Well, the power set is the set of all subsets of an infinite set.

Are all subsets of an infinite set finite?

.....

(Hint: Are there only a finite number or primes?)

(Hint 2: A set is a subset of itself. Is every infinite set finite?)

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For example, how would I be able to show that there exists a bijection between every element of the power set of the natural numbers and itself?

What's "itself"?

The elment itself? Are you asking if $X \in P(\mathbb N)$ does there exist a bijection $f: X\to X$?

All sets, no matter how the are defined, are bijective to themselves. Just use the identity function. $f(x) = x$ for all $x \in X$ is trivial to prove is bijective.

Or do you mean $P(\mathbb N)$ if $X\in P(\mathbb N)$ does there exist a bijection between $f:X \to P(\mathbb N)$?

The answer to that is no as Cantors diagonal shows.

Or did you mean a bijection between $f:X\to \mathbb N$? In that case only if $X$ is infinite.

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Is there a bijection between every element of that power set with the same cardinality?

SO if $X \in P(W)$ and $Y\in P(W)$ and $|X| = |Y|$ are you asking is there is a bijection between them?

Well, yes, that is the definition of having the same cardinality.