Let $ O=${$2n+1|n \in \Bbb N$} the odd natural numbers.

Show $|O|= \aleph_0$

I need to show that if there is a bijection $ \Bbb N \to O$, then $|O|=| \Bbb N|= \aleph_0$

Claim: $f$ is injective

Suppose $f(x)=f(x')$, $ \Rightarrow 2x+1=2x'+1 \Rightarrow 2x=2x' \Rightarrow x=x'$. So, $f$ is injective.

Claim: $f$ is surjective

$ \forall y \in O, \exists x=(y-1)/2 \in \Bbb N$ such that f(x)=y.

Then $f((y-1)/2)=2((y-1)/2)+1 \Rightarrow y-1+1=y$. Thus, $f$ is surjective.

Conclusion, $f$ is bijective. Thus, $| \Bbb N|=|O|=\aleph_0$.

If I have made any mistakes, please correct me. Thank you.

  • $\begingroup$ "do I ned to show that it is a bijective" really makes no sense, even adjusted for the misspelling. $\endgroup$
    – The Count
    Mar 21 '17 at 1:51

Here's a hint to get you started. To prove that two sets, $A$ and $B$, have equal cardinality (to show that $|A|=|B|$), we show that there exists a bijection $\phi: A\to B$. When dealing with $\aleph_n$, the typical procedure is to find a set $X$ with cardinality $\aleph_n$, and then prove the existence of a bijection from our set to $X$.

So, we know that $|\mathbb{N}|=\aleph_0$. Can you prove that there exists a bijection from $\mathbb{N}$ to $O$? You might try constructing one!

  • $\begingroup$ So, I need to show that $ | \Bbb N|=|O|$? $\endgroup$
    – Lily
    Mar 21 '17 at 1:55
  • $\begingroup$ Yes! Can you explain why? $\endgroup$ Mar 21 '17 at 1:56
  • $\begingroup$ If $| \Bbb N|=\aleph_0=|O|$. Then by its definition, there is a bijection from $ \Bbb N \to O$. $\endgroup$
    – Lily
    Mar 21 '17 at 2:02
  • 1
    $\begingroup$ True, but we want to work the other way -- if there is a bijection from $\mathbb{N}$ to $O$, then $|O|=|\mathbb{N}|$. Try coming up with this bijection, and post your proof as an answer! This way you can get feedback on your proof, too. $\endgroup$ Mar 21 '17 at 2:05
  • $\begingroup$ So, can I just say that my function $f(n)=2n+1$ and go from there? $\endgroup$
    – Lily
    Mar 21 '17 at 2:09

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