In this question given condition is : the selection operation have to distribute over theta join operation.
Where ii) says:
If all the attributes in the selection condition C involve only in the attributes of one of the relations being joined say, R
then two operations can be written as follows: σC( R ⋈ S ) ≡ (σC( R ) ) ⋈ R
so this option says that the selection operation is distribute over theta join operation . Which match with given condition.
Now for rest options :
- If the selection condition C can be written as C1 and C2, where condition C1 involves only the attributes of R and/or condition C2 involves only the attributes of S, then operations can be written as : either
i) σC (R ⋈ S) ≡ (σC1 (R)) ⋈ (σC2 (S))
or
iii) σC (R ⋈ S ) ≡ σC2 ( (σC1 (R) ) ⋈ S)
or
iv) σC (R ⋈ S ) ≡ σC1 ( R ⋈ (σC2 (S) ) )
all three options also support that given condition - " selection operation distribute over theta join operation " .
hence all Four options are correct .