False statements in the given options:

I. Relational algebra is more powerful than relational calculus.
II. Relational calculus is more powerful than relational algebra.
III. Relational algebra is as same power as relational calculus.

Explanation:

Relational algebra and relational calculus are two different approaches to query the relational database. They have different syntax and expressivity. Let's analyze each statement to determine if it is true or false.

I. Relational algebra is more powerful than relational calculus:

- This statement is false. Relational algebra is a procedural query language that uses a set of operations to manipulate relations. It provides a way to perform operations like selection, projection, join, union, etc. It is a formalism for reasoning about queries and designing queries.

- Relational calculus, on the other hand, is a non-procedural query language that specifies what data to retrieve from the database without specifying a particular procedure to obtain it. It uses logical formulas and quantifiers to express queries. It allows more complex and expressive queries.

- Relational calculus allows the specification of queries that cannot be expressed in relational algebra. For example, queries involving existential quantifiers or complex conditions cannot be expressed in relational algebra but can be expressed in relational calculus.

II. Relational calculus is more powerful than relational algebra:

- This statement is false. As explained above, relational calculus allows the specification of more complex and expressive queries compared to relational algebra. It allows the use of logical formulas and quantifiers to express queries.

- However, relational algebra is more powerful than relational calculus in terms of manipulation and optimization of queries. Relational algebra provides a set of operations that can be used to combine and transform relations effectively. It is a procedural language that allows reasoning about queries and designing efficient query execution plans.

- Relational algebra provides a foundation for query optimization techniques, such as query rewriting, join reordering, and selection pushing, which can significantly improve query performance. Relational calculus, being a non-procedural language, does not provide such optimization techniques.

III. Relational algebra is as same power as relational calculus:

- This statement is false. Relational algebra and relational calculus have different levels of expressivity. Relational calculus allows the specification of more complex and expressive queries compared to relational algebra.

- Relational calculus allows the use of logical formulas and quantifiers to express queries, which gives it a higher level of expressivity. Relational algebra, on the other hand, is more limited in terms of expressivity but provides a foundation for query manipulation and optimization.

Conclusion:

Based on the analysis, all the given statements are false. Relational calculus is more powerful than relational algebra in terms of expressivity, while relational algebra is more powerful in terms of query manipulation and optimization.