DATABASE SYSTEM CONCEPTS (LOOSELEAF)
DATABASE SYSTEM CONCEPTS (LOOSELEAF)
7th Edition
ISBN: 9781260515046
Author: SILBERSCHATZ
Publisher: MCG
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Chapter 16, Problem 2PE

a.

Explanation of Solution

To prove:

E1|><|θ(E2E3)E1|><|θE2E1|><|θE3

Proof:

  • To enhance the efficiency of E1|><| the queries, equivalences such as commutative, distributive, and associative properties can be applied.
  • In the given equivalence, distributive property is applied, which means the natural join is distributive over difference operation.
  • Let E1|><|θE2 as “R1” and E1|><|θE3 as “R2”.
  • If a record “r” belongs to left hand side (LHS) of the given equivalence, it will also be the member of “R1” part of right hand side (RHS) of the above equation

b.

Explanation of Solution

To prove:

σθ(AγF(E))(AγF(σθ(E))) where “θ” uses only attributes from “A”.

Proof:

  • It is given that, “θ” uses only attributes from “A”.
  • In the LHS of the equation, if  a record “r” in the result of AγF(E) is selected by the selection operation, then on the right hand side, all the records in relation “E” having value in “A” is equal to r[A] are selected by the selection operation and it is represented as,

  r,rσθ(AγF(E))r

c.

Explanation of Solution

To prove:

σθ (E1⟕E2) ≡ σθ (E1)⟕E2, where “θ” uses only attributes from “E1”.

Proof:

  • It is given that, “θ” uses only attributes from “A”.
  • In the LHS of the equation, if  a record “r” in the result of “E1⟕E2” is selected by the selection operation, then on the right hand side, all the records in relation “E1” having value is equal to r[E1] are selected by the selection operation and it is represented as,

r,rσθ(E1⟕E2)  rσθ <

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