What would be the equivalence class of the following be?
Let Z be a set of integers and n ∈ Z. Let R be the relation on Z by aRb if a - b is a multiple of n.
Denote the relation by a ≡ b (mod n).
Reflexive relation: For all a ∈ Z, a - a = 0 * n. a ≡ a (mod n) Symmetric relation: Let a,b ∈ Z such that a ≡ b (mod n). There is an integer k such that a -b = kn. Multiply both sides of the equality by (-1) and let k' = -k we find that b - a = k'n. b ≡ a (mod n)
Transitive Relation: Let a,b,c ∈ Z be such that a ≡ b (mod n) and b ≡ c ( mod n). There exist Z k1 and k2 such that a -b = k1n and b-c = k2n. Add the equalities together we find a - c = kn where k = k1 + k2 ∈ Z which shows that a ≡ c ( mod n)