$($ii$)$ Relational DBMSs process queries written in SQL$.$

a$.$ Use the generic relational algebra expression underpinning a typical SQL statement to argue why query optimization is needed when tables involved in the query contain a large number of rows.

b$.$ Natural join is a relational algebraic operator frequently used in queries. Estimating the size of the result table of a natural join operation helps optimizing queries. Assuming that natural join is performed between a table A of M rows and another table B of N rows, estimate the number of rows in the result table under three different situations: when the join attribute is the primary key of one of the table, when there is no common join attribute, and when the join attribute is not a primary key in either table.