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Proof of Set Relationships Using Element Definitions

A proof of set relationships using element definitions in mathematical set theory.

David Miller
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ProofofSetRelationshipsUsingElementDefinitionsQuestion:AN(BUC)=(ANB)U(ANC)Proof:Toshowthat4n(BUC)=(4nB)U(4NC),weprovein(1)thatAN(BUC)(ANB)U(ANC),andin(2)that(4nB)U(ANC)cAN(BUC).(1)An(BUC)c(ANB)U(ANC)WetakeanyelementxeAn(Bu).Bythedefinition,wehavethatxedandxeBuUC.AsxeBuC,wehavethatxeBorxeC.So,xe4andxeB,0rxe4andxeC.Namely,xeANBorxeAnC.Hence,xe(ANB)U(ANC).So,AN(BUC)c(ANB)U(ANC).

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