In a school of 180 students, each student belongs to exactly one of the four house colours: Red, Blue, Green, or Yellow. Define a relation H on the set of students by
PHQ⟺P and Q are in the same house colour.
Answer the following questions:
a. (i) Prove that H is an equivalence relation on the set of all students.
b. (ii) How many distinct equivalence classes does H produce?
c. (iii) If 45 students are in the Red house, determine the size of the equivalence class of any student who belongs to the Red house.