Solution for Section 10.3 Question 5

5. a) Recall the quotient - remainder theorem which states that given any integer n and positive integer d, there exist unique integers q and r such that
n = d · q + r  where 0r < d. Now let d = 5 and consider the possible values for r.

The quotient - remainder theorem guarantees that every integer x can be written in the form x = 5 · q + r  where 0r < 4.
The set A₀ includes all the integers for which r = 0.
The set A₁ includes all the integers for which r = 1.
The set A₂ includes all the integers for which r = 2.
The set A₃ includes all the integers for which r = 3.
The set A₄ includes all the integers for which r = 4.

The sets A₀, A₁, A₂, A₃ and A₄ are mutually disjoint sets.

Thus every integer occurs in precisely one of the sets A₀, A₁, A₂, A₃ and A₄, and hence these sets partition the set of integers.

b) To find the relation induced by this partition, compare the subsets A₀, A₁, A₂, A₃ and A₄ to the equivalence classes you found in the previous question.

The relation s induced by this partition is defined as follows:  for any integers m and n, m s n 5 | (m - n).

Back to Section 10.3