DBMS Relational Algebra

Question

How the following two expressions are equal?

The LHS will remove duplicates but RHS will not.Please explain

Comments

yes, these are equivalent only if they dont have any common attribute....

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Well common attribute will not be removed in this case.I am more concerned about the dupliate rows.As left one is type of natural join so it will use projection and remove duplicates but RHS will not be able to remove.I know i am missing something because these are equal under all cases

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read the statement written in bold....i have taken it directly from wikipedia.

If we want to combine tuples from two relations where the combination condition is not simply the equality of shared attributes then it is convenient to have a more general form of join operator, which is the θ-join (or theta-join). The θ-join is a binary operator that is written as {\displaystyle {R\ \bowtie \ S \atop a\ \theta \ b}} {\displaystyle {R\ \bowtie \ S \atop a\ \theta \ b}} or {\displaystyle {R\ \bowtie \ S \atop a\ \theta \ v}} {\displaystyle {R\ \bowtie \ S \atop a\ \theta \ v}} where a and b are attribute names, θ is a binary relational operator in the set {<, ≤, =, >, ≥}, v is a value constant, and R and S are relations. The result of this operation consists of all combinations of tuples in R and S that satisfy θ. The result of the θ-join is defined only if the headers of S and R are disjoint, that is, do not contain a common attribute.

The simulation of this operation in the fundamental operations is therefore as follows:

R ⋈θ S = σθ(R × S)
In case the operator θ is the equality operator (=) then this join is also called an equijoin

Answer 1

Both the expressions are equal.

The LHS is not natural join but conditional join, which is cross product of r and s and then projection of those tuples in the cross product that satisfy the given condition.

Consider r(A, B, C) and S(C, D)

r :

A	B	C
1	2	3
2	1	2

and s :

C	D
3	4
1	4

and let the condition c be r.B < s.C 

So, r $\bowtie$_c s will be :

r.A	r.B	r.C	s.C	s.D
1	2	3	3	4
2	1	2	3	4

An important point to note here is that unlike in natural join, conditional join will contain all the columns of cross product.

And r X s will be :

r.A	r.B	r.C	s.A	s.B
1	2	3	3	4
1	2	3	1	4
2	1	2	3	4
2	1	2	1	4

And after applying the condition r.B < s.C, we get

r.A	r.B	r.C	s.C	s.D
1	2	3	3	4
2	1	2	3	4

Comments

@rahul sharma 5

 Natural join removes duplicates . We can perform a Natural Join only if there is at least one common attribute that exists between two relations.

LHS is Conditional join.

And RHS is Cartesian product . That combines information of two different relations into one relation .

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In question https://gateoverflow.in/136247/dbms-number-of-tuples-in-ra#a136312 for the first part.When we are taking thetha join we have common attribute.Is this wrong then?As nitish is saying that there should be no common attribute in thetha join?

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From wikipedia

The simulation of this operation in the fundamental operations is therefore as follows:

R ⋈_θ S = σ_θ(R × S)

Answer 2

R intersection S = § then R  NATURAL JOIN S=R×S

Answer 3

The LHS is a theta Join not a natural join, so it won't remove duplicates and the RHS is a condition enforced on Cartesian Product. 

In Conditional Join  AKA Theta Join, while performing cartesian product only the condition is evaluated.
In the RHS, first the Cartesian Product is evaluated and then the condition is enforced.

Both result in same output.