 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/3/e/ii/solution

This is equality of remainders in modular arithmetic. It is reflexive because

3 ∣ (n - n) = 0

, symmetric because

3 ∣ (n - m)

implies

3 ∣ (m - n)

, and transitive because

3 ∣ (n - m)

and

3 ∣ (k - n)

imply

3 ∣ (k - m)

by addition. Thus

m \sim n ⟺ m \equiv n (mod 3) is an equivalence relation .

C_{r} = {n \in N : n \equiv r (mod 3)}, r = 0, 1, 2.

If zero is included in

N

, it belongs to

C_{0}

; otherwise that class starts at three.

 New to topics? Read the docs here!