Give examples to show that a Projection cannot be pushed be

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Give examples to show that: a) Projection cannot be pushed below set union. b) Projection cannot be pushed below set or bag difference. c) Duplicate elimination (delta) cannot be pushed below projection. d) Duplicate elimination cannot be pushed below bag union or difference.

Solution

a)

A(a,b) = {(3,4)}

B(a,b) = {(3,5)}

a (A B) = {(3),(3)}

aA aB = {(3)}

b)

For Sets: A(a,b) = {(3,4)}

B(a,b) = {(3,5)}

a(A-B) = {(3)}

(Aa Ba) =

For Bags: A(a,b) = {(3,4),(3,4),(3,4)}

B(a,b) = {(3,4), (3,5)}

a(A-B) = {(3),(3)}

(Aa Ba) = {(3)}

c)

A(a,b) = {(3,4),(3,5)}

a((A)) = {(3),(3)}

((Aa)) = {(3)}

d)

A(a,b) = {(3,4),(3,4),(3,4)}

B(a,b) = {(3,4), (3,5)}

(A B) = {(3,4),(3,4),(3,4),(3,4), (3,5)}

(A) (B) = {(3,4),(3,4), (3,5)}