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## Result of a set difference operator in relational algebra

#### Question:

6. Which of the following is equivalent to the expression R - (R - S)?

(a) R
(b) S
(c) R - S
(d) R ∩ S

(d) R ∩ S

R - (R - S) is equivalent to R ∩ S . This could be understood from the following example;

Let us take the following tables (relations) R and S with 3 and 2 tuples respectively.
 R B D Tuple 1 b1 d1 Tuple 2 b2 d2 Tuple 3 b3 d3

 S B D Tuple 1 b1 d4 Tuple 2 b3 d3

The following table shows R-S. The result contains all the records that are unique for R and not for S. In other words, the common tuple in R and S is removed and the tuples that are only part of R is retained.

 R – S B D Tuple 1 b1 d1 Tuple 2 b2 d2

The following table shows R-(R-S). The result contains records that are unique for R and not for (R-S).

 R – (R – S) B D Tuple 1 b3 d3

From the result, it is very clear that the tuple (b3, d3) is the common tuple that presents in both R and S. Hence, R-(R-S) is equivalent to R ∩ S.