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I need to prove that these two sets have the same cardinality: $$\mathbb{R}^\mathbb{N}$$ and:

$$\mathcal{P}( \mathbb{N})$$

I thought about using these known facts:
If $A,B,C$ sets then:
1. $A^{B^ {~C}} \sim A^{B \times C}$
2. If $B$ and $C$ are disjoint then $A^B \times A^C \sim A^{B~ \cup ~C}$

But I am stuck on that proof.. I tried to set:
$A = \mathbb{R}$ , $B = \mathbb{N}$ , and $C = \{0\}$ and to use the known facts above while using $|A^B| = |A|^{|B|}$

I would appreciate your help, thank you!

Asaf Karagila
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CSch of x
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  • $\mathbb{R}^\mathbb{N}\sim\mathbb{R}\sim 2^\mathbb{N}$ – Chrystomath May 29 '20 at 16:04
  • @Chrystomath hey thank you for answering! I don't recognize this $\approx$ symbol. is it meant to be a tilde ? ( $ \sim $) thank you again – CSch of x May 29 '20 at 16:05
  • It helps a lot if your titles are descriptive and tells us what is in the question, rather than it is a question. – Asaf Karagila May 29 '20 at 16:32
  • @AsafKaragila Sorry I cant seem to find help or hints to the solution for this question, I will edit the title to be more informative – CSch of x May 29 '20 at 17:38
  • I have put three different questions that in one way or another combine to an incredibly complete solution of this problem. – Asaf Karagila May 29 '20 at 17:39
  • @AsafKaragila I will look deeper. Thank you! – CSch of x May 29 '20 at 17:40
  • @AsafKaragila Sorry, I still do not understand how to solve it, you gave me links but in the links themselves there aren't any full solutions that I can be based on, I can't solve this question nor the the parts where the answers said "I will leave it to you to prove it" at https://math.stackexchange.com/questions/243590/bijection-from-mathbb-r-to-mathbb-rn – CSch of x May 30 '20 at 14:58
  • Can you use the first part of my answer in the page you just linked to? – Asaf Karagila May 30 '20 at 15:31
  • @AsafKaragila If by the first part you meen that $\mathbb{(N^N)^N\sim N^{N\times N}\sim N^N}$ I can only use the first isomorphism, not the second (the first one is fact #1 in this post) – CSch of x May 30 '20 at 15:34
  • And you can't figure out any way and form that this can be helpful to your question? – Asaf Karagila May 30 '20 at 17:13
  • @AsafKaragila I've tried it here: https://math.stackexchange.com/questions/3698465/proof-checking-mathbbr-mathbbn-sim-mathcalp-mathbbn-discr/3698479#3698479 – CSch of x May 30 '20 at 17:24

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