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I know that two sets are equinumerous if there exist a bijection between them, and they are uncountables if there exist another bijection between the real numbers from 0 to 1 and a set.

So, as they requiere both conditions, can I maintain that they are equinumerous?

AMB
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No you cannot prove this statement because it is false. Cantor showed that the power set of a set is strictly larger than the set. The reals are uncountable and the power set of the reals is strictly larger, so these two sets are not equinumerous. In fact there is a huge number of uncountable cardinalities.

Ross Millikan
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  • https://math.stackexchange.com/questions/1226677/proving-intervals-are-equinumerous-to-mathbb-r I found this, are intervals equinumerous to R but power set not? – AMB Oct 08 '17 at 15:01
  • That is correct. You can find simple bijections between open intervals an all of $\Bbb R$, like by using the arctangent. – Ross Millikan Oct 08 '17 at 18:58
  • So all intervals of real numbers are equinumerous but they are not equinumerous with their power sets? E.g. the set of all the real numbers is not equinumerous with the power set of the reals? – AMB Oct 08 '17 at 21:20
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    That is correct. No set is equinumerous with its power set. The cardinal of the reals is denoted by $\mathfrak c$, the power of the continuum and is equinumerous with the power set of the naturals. – Ross Millikan Oct 08 '17 at 22:57