On 04.12.2012 11:01, William Elliot wrote: > On Mon, 3 Dec 2012, jaakov wrote: > >> Given a set X and a cardinal k, is there a set Y such that card(Y)=k and X is >> disjoint from Y? > > > Case |X|< k. Let Y be a set with |Y| = k. > |Y\X| = k; Y\X and X are disjoint. > Case |X|<= k. Let A be a set with k< |A|. k< |A| = |A\X|. > Take Y as any subset of A\X with |Y| = k. > >> Is there a proof of this fact that works without the axiom of regularity (= >> axiom of foundation) and does not assume purity of sets?