Consider the relational database defined below:

create table customer ( customer_ID varchar ( 1 0 ) ,

customer_name varchar ( 3 0 ) ,

ZIP_code char ( 5 ) , primary key ( customer_ID ) ) ;

create table product ( product_name varchar ( 1 0 ) ,

price numeric ( 1 0 , 2 ) , primary key ( product_name ) ) ;

create table purchase ( purchase_ID varchar ( 1 0 ) ,

customer_ID varchar ( 1 0 ) ,

product_name varchar ( 1 0 ) ,

quantity int , primary key ( purchase_ID ) ,

foreign key ( customer_ID ) references customer ,

foreign key ( product_name ) references product ) ;

Express in relational algebra each of the following queries:

(a) (10 points) For each ZIP code stored in the customer table, find the number of customers who have purchased product X.

(b) (10 points) For each product whose price is less than $10,000, increase the price of that product by 10 percent. For all other products, increase their prices by 5 percent.

(c) (10 points) Find the names of the products that have not been purchased by anyone whose ZIP code is 12222.