For each of the following pairs of expressions, give instances of relations that show the expressions are not equivalent.

a. Π_A(R − S) and Π _A (R) – Π_A (S)

b. σ_{B < 4}(AG_max(B)(R)) and AG_max(B)(σ_B<4(R))

c. In the preceding expressions, if both occurrences of max were replaced by min would the expressions be equivalent?

d. (R Θ S) Θ T and R Θ (S Θ T)

In other words, the natural left outer join is not associative.

e. σ_θ (E₁ Θ E₂) and E₁ Θ σ_θ (E₂), where θ uses only attributes from E₂