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 LauSep 15, 2017 at 5:28
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$\begingroup$ @KennyLau ZFC set theory. $\endgroup$– user107952Sep 15, 2017 at 5:29
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3$\begingroup$ Looks like a duplicate of this question or this one. $\endgroup$– bofSep 15, 2017 at 5:36
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$\begingroup$ 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 ElliotSep 15, 2017 at 11:23
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1 Answer
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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.
