Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the cardinality of $\Bbb{N^N}$?

my answer: $|\mathbb{R}|$ $=$$|2^\mathbb{N}|$ $\leqslant$ $|\mathbb{N}^\mathbb{N}|$ $\leqslant$ $|\mathbb{R}^\mathbb{N}|$ $=$ $|(2^\mathbb{N})^\mathbb{N}|$ $=$ $|2^{\mathbb{N}\times\ \mathbb{N}}|$ $=$ $|2^\mathbb{N}|$ $=$ $|\mathbb{R}|$

In the end $|\mathbb{N}^\mathbb{N}|$ $=$ $|\mathbb{R}|$

Is that okay?

share|improve this question
2  
Yes, except that the third item from the end should be $\left|2^{\Bbb N\times\Bbb N}\right|$. –  Brian M. Scott Jan 20 '13 at 20:24
    
Yes, but i didn't know \times\. Thank you –  Agenog Jan 20 '13 at 20:27
    
You’re welcome. –  Brian M. Scott Jan 20 '13 at 20:27

2 Answers 2

Yes. The answer is fine.

(Incidentally my students just have this question in their homework)

share|improve this answer
2  
Maybe he is your student ;) –  Belgi Jan 20 '13 at 20:52
    
I doubt that, we gave a slightly different formulation. –  Asaf Karagila Jan 20 '13 at 21:04
    
I was kidding, though I guess you gave your homework in Hebrew so the different formulation can be a result of translation –  Belgi Jan 20 '13 at 21:06
    
Oh, it was more than that. It was different mathematically :-) –  Asaf Karagila Jan 20 '13 at 21:13

Your answer is correct, though I would of added that you know that for cardinals $k\leq l,l\leq k\,\implies k=l$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.