I was going about this by trying to come up with a bijection between this set and the set of all infinite sequences of zeros and ones (like 010100....), which I know in uncountable. But I didn't have much success. Other thoughts on how to get started?

  • What is a complete relation? – bof Sep 14 '16 at 4:54
  • For any x and y, either xRy or yRx or both – bitter-sweet Sep 14 '16 at 14:34
  • So complete implies reflexive? or is it only for distinct x and y? – bof Sep 14 '16 at 20:52

It is enough to come up with an injection from the set of infinite sequences of zeros and ones into the set of complete and transitive relations.

So suppose you have a sequence like $(a_n)=(0,1,0,1,0,0,\ldots)$. You want to define a relation $R$ that depends somehow on $(a_n)$. You're going to say that $iRj$ if and only if... [something to do with $a_i$ and $a_j$]. Can you come up with a definition that makes $R$ complete and transitive? And is the mapping from sequences to relations injective?

  • I thought about the conditions ai>aj, aj>=aj, ai=aj, ai=aj=0, ai=aj=1. None of them work. Can you please give a further hint? – bitter-sweet Sep 14 '16 at 14:38
  • Which condition came closest to working? Is it close enough, or is the problem small enough to work around? – Chris Culter Sep 14 '16 at 17:29
  • Ok how about I keep the condition ai=aj=0, except that for sequences with ak=0 for exactly one value of k, I take the corresponding relation to be kRw for all w, and not mRn for any other m,n? – bitter-sweet Sep 14 '16 at 19:32
  • Well, if the condition requires $a_i=a_j$, then whenever $a_i\neq a_j$, we have that $i$ and $j$ are not related by $R$, which means $R$ isn't complete. What about the other conditions? – Chris Culter Sep 14 '16 at 21:11
  • Can't believe I missed that. Then ai>=aj is the candidate condition. This gives complete & transitive relations, but it isn't an injection because for eg. the relation corresponding to 'all zeros' and 'all ones' is the same.. – bitter-sweet Sep 14 '16 at 22:23

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