Transcription of Identity 2. - gatech.edu
1 Identity (B A) =A (B A) =A (B Ac)set difference=A (Ac B)commutative= (A Ac) (A B)distributive=U (A B)complement=A Bidentity A (B A). Thenx Aorx (B A) by definition ofunion. Sox Bandx6 A(by set difference). Butx Aby previousstatement, sox Aorx B. By definition of union,x (A B). Identity (A Bc)c B=Ac BProof.(A Bc)c B= (Ac (Bc)c) Bde Morgan s= (Ac B) Bdouble complement=Ac (B B)associative=Ac Bidempotent Identity , (A B) C=A (B C)Proof.(A B) C= (A Bc) Cset difference= (A Bc) Ccset difference=A (Bc Cc)associative=A (B C)cde Morgan s=A (B C)set difference (A B) C. Thenx (A B) andx6 Cby definition ofset difference.
2 Further,x Aandx6 Balso by definition of set Aandx6 Bandx6 C, which impliesx6 (BorC). Hence,x6 (B C) by definition of union. Thus, givenx Awe havex A (B C)by definition of set difference. 1 Identity , (B A) (C A) = (B C) AProof.(B A) (C A) = (B Ac) (C Ac)set difference= (B C) Acdistibutive= (B C) Aset difference (B A) (C A). Thenx (B A) orx (C A).Ifx (B A), thenx Bandx6 Aby definition of set difference. Ifx (C A), thenx Candx6 Aby definition of set difference. Thus,x Borx Candx6 A. By definition of union,x (B C). By definitionof set difference, ifx (B C) andx6 Athenx (B C) A.
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