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 :
and s :
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 |