# Explicit set-theoretic construction of a disjoint set with same cardinality

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

• Under which set theory? – Kenny Lau Sep 15 '17 at 5:28
• @KennyLau ZFC set theory. – user107952 Sep 15 '17 at 5:29
• Looks like a duplicate of this question or this one. – bof Sep 15 '17 at 5:36
• Is S a subset of S'? If so, then for S' to be disjoint and equinumerous to S, both have to be empty. – William Elliot Sep 15 '17 at 11:23