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So i have

Sℕ = P(ℕ)

1. I need 3 elements of Sℕ that are finite sets. My idea:

Sℕ = {x | x < 4)

does that work?

2. I need to write down an element of Sℕ that is an infinite set.

Well, ℕ itself is already infinite ℕ=(1, 2,3,4,5......n)

So my answer would be:

Sℕ = {x | x €  ℕ}

Correct? If not, would you like to correct me?

Thanks in advance!

Prometheus
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  • What does $S\mathbb N$ mean? – lulu Oct 21 '20 at 12:57
  • Sℕ = P(ℕ) so S contains all power sets of ℕ. – Prometheus Oct 21 '20 at 12:59
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    Not following. Why introduce a new notation for the power set? Are you just asking for three finite subsets of the natural numbers? – lulu Oct 21 '20 at 13:00
  • I am going to ignore the fact that I ever saw $S\Bbb N$... and will replace every occurrence of $S\Bbb N$ with $\mathcal{P}(\Bbb N)$ in my mind and response... I'm with lulu on this. It should never have been written that way. That said, if you are looking for three elements of $\mathcal{P}(\Bbb N)$ which are finite sets, then one of the elements could be ${x\in\Bbb N\mid x<4}$ which is otherwise more simply written as ${1,2,3}$. You still need two more elements... so how about ${1,2,3,4}$ and ${1,2,3,4,5}$. You only wrote one so far and it sounds like you thought this was three. – JMoravitz Oct 21 '20 at 13:16
  • Next, you wrote "ℕ=(1, 2,3,4,5......n)" No... $\Bbb N = {1,2,3,4,5\dots}$. You wrote it with an $n$ at the end which makes it sound as though you think it stops at some point at some number $n$. The set ${1,2,3,\dots,n}$ is the $n$-element set ${x\in\Bbb N\mid 1\leq x\leq n}$ otherwise written as $[n]$. As for an infinite element of $\mathcal{P}(\Bbb N)$, yes $\Bbb N$ is indeed an infinite element of $\mathcal{P}(\Bbb N)$. So too is the set of prime numbers ${2,3,5,7,11,13,17,\dots}$ and the set of even numbers ${2,4,6,8,10,12,14,\dots}$ and many many many more. – JMoravitz Oct 21 '20 at 13:19
  • Finally... please learn to use MathJax so you can correctly type things like $x\in\Bbb N$ rather than x € ℕ. There is no excuse for using the symbol for a Euro rather than the $\in$ symbol. – JMoravitz Oct 21 '20 at 13:21
  • thanks! However, P(ℕ) can't exist inside a power set. So i assume my solution simply saying P(ℕ) = {x | x ∈ ℕ} can't be quite right? – Prometheus Oct 21 '20 at 13:41
  • Please edit your post for clarity. What do you mean by $P(\mathbb N)$? I assumed you meant the power set of the natural numbers. That is, the set of subsets of the natural numbers. But you say that "$P(\mathbb N)$ can't exist inside a power set" which is hard to interpret. – lulu Oct 21 '20 at 13:53
  • Oh, certainly, you are using equal signs in places you absolutely shouldn't. An element of $\mathcal{P}(\Bbb N)$ which is infinite is $\Bbb N$ itself. This is not the same as the statement that $\mathcal{P}(\Bbb N)$ is equal to $\Bbb N$ which is clearly wrong. You should have written instead as $\mathcal{P}(\Bbb N)\ni {x\mid x\in\Bbb N}$ using $\ni$ instead of $=$. As for "$\mathcal{P}(\Bbb N)$ can't exist inside a power set" No, it absolutely can and exists as an element of $\mathcal{P}(\mathcal{P}(\Bbb N))$ for instance, the powerset of the powerset of $\Bbb N$. – JMoravitz Oct 21 '20 at 13:59

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