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I am trying to show that $|\Bbb {R} \times \Bbb {R}| \leq |\Bbb {R}|$.

I don't know how to define $f:\Bbb {R} \times\Bbb {R} \rightarrow \Bbb {R}$ in a way that would make $f$ injective.

My professor gave us the hint that $(0,1) \sim \Bbb {R}$ would imply that $|(0,1) \times (0,1)| \leq |(0,1)|$, but I don't understand how that is helpful here.

user45417
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  • Is $\mathbb{R}$ isomorphic to the interval $(0,1)$? – NoName Sep 21 '14 at 19:51
  • Again, it might be worth to point out that this is your third question on the topic. In particular, the previous one was closed as a duplicate. Why is this not another duplicate (since you're essentially asking the same question as before)? – Asaf Karagila Sep 21 '14 at 20:01
  • (And I don't like that I keep pointing you to threads that already exist, and you keep ignoring those referrals.) – Asaf Karagila Sep 21 '14 at 20:03
  • @AsafKaragila These referrals are of no help whatsoever. None of the other answers to my questions were particularly helpful and I am just as confused on how to do the problem as I was the last time that I asked a question about it. No ever fully answered my question. This specific part of my question in particular. I am merely trying to understand how to do the problem and have not yet been able to. – user45417 Sep 21 '14 at 20:29
  • (1) You should point that out. Not just repeat the same question over and over again. Einstein said that insanity is doing the same thing over and over again, expecting different outcome. (2) There are at least two answers which fully detail an injection from $(0,1)^2$ into $(0,1)$ in those duplicates. So when you do add details, please include "I have read this and that answer, and I don't understand because ...". – Asaf Karagila Sep 21 '14 at 20:36

1 Answers1

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Consider, for items in $(0, 1)$, writing a decimal form of two numbers $a = 0.a_1 a_2 a_3 \ldots$ and $b = 0.b_1 b_2 b_3 \ldots$ with the rule that you never write "$...d999\ldots$", but instead write $...(d+1) 0000$. Then you can write the number $0.a_1 b_1 a_2 b_2 \ldots$...

To illustrate with a couple of examples:

  1. Rather than write $0.2$ as $0.199999\ldots$, you write $0.2$.

  2. If your numbers are $a = 0.12340000000\ldots$ and $b = 0.5712121212\ldots$, then the number they map to is $$ 0.15273142010201020102\ldots; $$ its digits are those of $a$ interleaved with those of $b$.

Given @Asaf's comments, I think I'm done explaining here.

John Hughes
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  • So would that mean to define $f: \Bbb {R} \times \Bbb {R} \rightarrow \Bbb {R}$ as $f((d+1)0000.)=0.a_1 b_1 a_2 b_2...$ ? – user45417 Sep 21 '14 at 19:57
  • No. See my extended answer. If you've been wondering about this question for four days (and looking at your earlier questions, I'm seeing some pretty fundamental misconceptions), it's possible that you're in the wrong course, or trying to do self-study from the wrong book. It appears that this question, even when fully answered, is beyond you; that should be a message to you to try to better understand the material that comes before it. – John Hughes Sep 21 '14 at 21:14