Correct Answer - Option 4 : Only I), III) and IV)
Statement I : ΠP (R ⋈ S)
In I, Ps from natural join of R and S are selected.
Statement II : ΠP (R) ⋈ ΠP (S)
II is not equivalent as it may also include Ps where Qs are not same in R and S.
Statement III : ΠP (ΠP, Q (R) ∩ ΠP, Q (S))
All Ps from the intersection of (P, Q) pairs present in R and S.
Statement IV ΠP (ΠP, Q (R) - (ΠP, Q (R) - ΠP, Q (S)))
It is also equivalent to III because of (R – (R – S)) = R ∩ S.
Hence the Option I is the correct answer
Natural Join: NATURAL JOIN is a JOIN operation that creates an implicit join clause for you based on the common columns in the two tables being joined. Common columns are columns that have the same name in both tables. A NATURAL JOIN can be an INNER join, a LEFT OUTER join, or a RIGHT OUTER join.
Intersection: The intersection operator gives the common data values between the two data sets that are intersected. The two data sets that are intersected should be similar for the intersection operator to work. Intersection also removes all duplicates before displaying the result