Which of the following statements regarding relational algebra are true?
(i) R \bowtie_{<condition>} S \equiv \sigma_{<condition>} (R \times S)
(ii) R \bowtie_{<condition>} S \equiv \Pi_{<condition>} (R \times S)
(iii) R \bowtie_{<condition>} S \equiv R * \rho_{<list} S
Explanation
In relational algebra, the Theta Join (\bowtie_{<condition>}) is a fundamental operation used to combine tuples from two relations based on a specific condition.
Statement (i) is TRUE: A theta join is defined as a cross product (Cartesian product, ×) followed by a selection (σ) operation that filters the resulting tuples based on the specified condition.
\text{Theta Join Identity} \implies R \bowtie_{<condition>} S = \sigma_{<condition>} (R \times S)
Statement (ii) is FALSE: The symbol Π represents the Projection operation, which is used to select specific columns from a relation, not to join tables based on conditions.
Statement (iii) is FALSE: The symbol ρ represents the Rename operation, which is used to rename attributes or relations, not to perform a join.
To visualize the equivalence in statement (i):
\sigma_{<condition>} (R \times S) \equiv \text{Cartesian Product followed by filter}
By applying the filter to the Cartesian product, we effectively extract only those rows where the join condition holds true, which is the definition of a join.
