Result of set difference operator in relational algebra

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

Answer:

(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.

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