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 RS. 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(RS). The result contains records that are unique for
R and not for (RS).
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(RS) is equivalent to R ∩ S.