For each of the statements below, state whether it is true or false and then prove your answer.

(a) 15n 3 log n + 10n 2 + 50 is O(n 3 log n).

(b) 3n 2 − 12n + 2 is Ω(n 3 )

(c) 2n+1 is Θ(2n )

(d) 22n is O(2n )

(e) log(n!) is O(n log n) (Hint: compare n! and n n )

8. create an abstract student class for parker university. the class contains fields for student id number, last name, and annual tuition. include a constructor that requires parameters for the id number and name. include get and set methods for each field; the settuition() method is abstract. create three student subclasses named undergraduatestudent, graduatestudent, and studentatlarge, each with a unique settuition() method. tuition for an undergraduatestudent is $4,000 per semester, tuition for a graduatestudent is $6,000 per semester, and tuition for a studentatlarge is $2,000 per semester. write an application that creates an array of at least six objects to demonstrate how the methods work for objects for each student type. save the files as student.java, undergraduatestudent.java, graduatestudent.java, studentatlarge.java, and studentdemo.java.