Question: Problem (Proving set identities) For any two sets S and T, we define the symmetric difference of S and T, denoted by SAT and defined

Problem (Proving set identities) For any two sets S and T,

Problem (Proving set identities) For any two sets S and T, we define the symmetric difference of S and T, denoted by SAT and defined by SAT=(S\T)U(T\S). Prove each of the following statements, using only the set identities and proof techniques studied in the lectures. 1. The operation A is associative, that is, for any three sets A,B,C, we have: AA(BAC)=(AAB)AC. 2. For any three sets A,B,C, we have: AUBUC=(ANBNC)U(ANBNCJu(BNCNAJU(COBNAJU(ANBNCJu(BNCHAJU(CHANB) 3. For any three sets A,B,C, we have: AA(BAC)= AUBUC=((ANB)\C)U(ANC) \B)U(BNC) \A). Problem (Proving set identities) For any two sets S and T, we define the symmetric difference of S and T, denoted by SAT and defined by SAT=(S\T)U(T\S). Prove each of the following statements, using only the set identities and proof techniques studied in the lectures. 1. The operation A is associative, that is, for any three sets A,B,C, we have: AA(BAC)=(AAB)AC. 2. For any three sets A,B,C, we have: AUBUC=(ANBNC)U(ANBNCJu(BNCNAJU(COBNAJU(ANBNCJu(BNCHAJU(CHANB) 3. For any three sets A,B,C, we have: AA(BAC)= AUBUC=((ANB)\C)U(ANC) \B)U(BNC) \A)

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