Transcription of Identity 2. - gatech.edu
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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.
Further, x ∈ A and x 6∈B also by definition of set difference. Thus x ∈ A and x 6∈B and x 6∈C, which implies x 6∈(BorC). Hence, x 6∈(B∪C) by definition of union. Thus, given x ∈ A we have x ∈ A−(B∪C) by definition of set difference. 1. Identity 4. Let A, B and C be sets. Show that
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