Relational Algebra and Optimization Query optimizers make use of equivalences between queries. Two queries are equivalent Q1 = Q2 if for all instances of the relations used in the queries, R, the result of Qı on R is identical to the result of Q2 on R. Most optimizers make use of general equivalences that hold for all possible instances of the relations regardless of any integrity constraints. However, in the presence of integrity constraints, there may be additional equivalences that hold on the set of valid instances of the relations. Within semantic query optimizers, additional transformations are used that hold for the specified set of integrity constraints. Given relations Q[A, B] (A is key), R[A, B] (A is key), S[B, C) and T[B, C] (B is key for S and T). Prove or disprove the following equivalence: (Q – R) (S – T) = ( Q S ) – ( R T ). Hint: start by considering whether you can disprove this equivalence for relations that do not satisfy the given key constraints. Hint: to proved equivalence of Q1 = Q2 you can prove that Q1 C Q2 and Q2 C Q1. Show transcribed image text Relational Algebra and Optimization Query optimizers make use of equivalences between queries. Two queries are equivalent Q1 = Q2 if for all instances of the relations used in the queries, R, the result of Qı on R is identical to the result of Q2 on R. Most optimizers make use of general equivalences that hold for all possible instances of the relations regardless of any integrity constraints. However, in the presence of integrity constraints, there may be additional equivalences that hold on the set of valid instances of the relations. Within semantic query optimizers, additional transformations are used that hold for the specified set of integrity constraints. Given relations Q[A, B] (A is key), R[A, B] (A is key), S[B, C) and T[B, C] (B is key for S and T). Prove or disprove the following equivalence: (Q – R) (S – T) = ( Q S ) – ( R T ). Hint: start by considering whether you can disprove this equivalence for relations that do not satisfy the given key constraints. Hint: to proved equivalence of Q1 = Q2 you can prove that Q1 C Q2 and Q2 C Q1.
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