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Under what condition does the selection operation distribute over theta join operation?

Let $R$ and $S$ are two relations then which one of the following statements about the above condition is TRUE?

  1. $\sigma C (R \bowtie S) \equiv  (\sigma C1 (R)) \bowtie (\sigma C2 (S))$
  2. $\sigma C (R \bowtie S) \equiv (\sigma C (R)) \bowtie S$
  3. $\sigma C (R \bowtie S ) \equiv \sigma C2 ( (\sigma C1 (R)) \bowtie S)$
  4. $\sigma C (R \bowtie S ) \equiv  \sigma C1 (  R \bowtie (\sigma C2  (S)))$
  1. i, iii
  2. i, ii, iii, iv
  3. ii, iii
  4. i, ii
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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 .

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