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Q.1. Does theta join operator requires following union compatibility requirements?:

1. Same number of columns
2. Domain of corresponding columns should be same

I feel no, since I came across following fact:

$\sigma_\theta( R_1\times R_2)=R_1⋈_\theta R_2$

But I need confirmation.

Q.2. Since theta join can be represented in terms of cartesian product and selection as shown above, does it mean columns of resultant relation will contain ALL columns from both $R_1$ and $R_2$?

not getting ur point

Can u show ur doubt with some example

though I think no will be ans
$\theta$ join and conditional join are same except in $\theta$ join condition should be among attributes like R.a>S.b can't be like R.a>7.so there is no required to be union compatible.