Question

Given the below relational algebra expressions, use domain and tuple relational calculus to specify them:

a. σ_x=z ( R(a,b,c) )

b. π_x,y ( R(x,y,z) )

c. R(x, y) / S(x)

d. R(a,b,c) ∪  S(a,b,c)

e. R(a,b,c) – S(a,b,c)

f. R(d,e,f) ∩ S(d,e,f)

g. R(x,y,z) × S(f,g,t)

Answer 1

In Data Base Management System:

(a) σ_x=z ( R(a,b,c) ):

is the select operator, where:

Therefore, σ_x=z ( R(a,b,c) ) implies:

Select the tuple R(a,b,c) where x=z

(b) π_x,y ( R(x,y,z) ):

is the project operator, where:

Therefore π_x,y ( R(x,y,z) ) implies:

From a database take the columns x and y

(c) R(x, y) / S(x):

The division operator works as follows:

R(Z) / S(X), where . Let Y = Z - X (and hence Z
= X Y); that is, let Y be the set of attributes of R that are
not attributes of S.

Therefore R(x,y)/S(x) implies:

Where, T<-R(A,B)/S(A)

(d)R(a,b,c) ∪  S(a,b,c):

The result of , is a relation that includes all tuples that are either in R or in S or in both R and S

Duplicate tuples are eliminated

(e)R(a,b,c) - S(a,b,c):

The result of , is a relation that includes all tuples that are in R but not in S.

The attribute names in the result will be the same as the attribute names in R.

(f) R(d,e,f) ∩ S(d,e,f):

The result of the operation , is a relation that includes all tuples that are in both R and S

The attribute names in the result will be the same as the attribute names in R

(g) R(x,y,z) × S(f,g,t):

This operation is used to combine tuples from two relations in a combinatorial fashion.

Denoted by R(A1, A2, . . ., An) x S(B1, B2, . . ., Bm), The result is a relation Q with degree n + m attributes: Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.

Hence, if R has tuples (denoted as |R| = ), and S has tuples, then R x S will have tuples.