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Is there a set-theoretic construction, given an arbitrary set $S$, of a set $S'$ that is equicardinal with $S$ but is disjoint from $S$?

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  • $\begingroup$ Under which set theory? $\endgroup$ – Kenny Lau Sep 15 '17 at 5:28
  • $\begingroup$ @KennyLau ZFC set theory. $\endgroup$ – user107952 Sep 15 '17 at 5:29
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    $\begingroup$ Looks like a duplicate of this question or this one. $\endgroup$ – bof Sep 15 '17 at 5:36
  • $\begingroup$ Is S a subset of S'? If so, then for S' to be disjoint and equinumerous to S, both have to be empty. $\endgroup$ – William Elliot Sep 15 '17 at 11:23
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Here's my attempt. Consider O to be an ordinal whose cardinality is larger ,than that of any element of S. Then take S' to be the set of disjoint unions of elements of S with O. There is an obvious bijection between S and S' and these two sets must be disjoint due to their elements having distinct cardinals.

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