To show that the relation is indeed an equivalence relation we have to check that the relation is
reflexive
symmetric
transitive
a = a mod I,
since a - a = 0
and so belongs to I.
Suppose a = b mod I. Then a - b
belongs to I and hence so does
-(a - b) = b - a. But this means
b = a mod I.
Suppose
a = b mod I and b = c mod
I. Then a - b and b - c belong to
I and then so does their sum (a - b) + (b
- c) = a - c. But this means that a =
c mod I.