# Question

Show that the following equivalences hold. Explain how you can apply then to improve the efficiency of certain queries:

a. E1 Θθ (E2 − E3) = (E1 Θθ E2 − E1 Θθ E3).

b. σθ (AGF (E)) = AGF (σθ (E)), where θ uses only attributes from A.

c. σθ (E1 Θ E2) = σθ(E1) Θ E2 where θ uses only attributes from E1.

a. E1 Θθ (E2 − E3) = (E1 Θθ E2 − E1 Θθ E3).

b. σθ (AGF (E)) = AGF (σθ (E)), where θ uses only attributes from A.

c. σθ (E1 Θ E2) = σθ(E1) Θ E2 where θ uses only attributes from E1.

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