What Are the Equivalence Classes of the Smallest Relation Containing R?

[jacktsoi]

Homework Statement

Let R denote the following relation in the set N of natural numbers:
R = {(x, y) | x = 2y}.
Let E be the “smallest” equivalence relation containing R. Give a complete description of the equivalence classes of E.

How can I describe it containing infinite set?

Homework Equations

[An example of smallest equivalence relation can be given as follows::
Let S = {a, b, c, d} and R = {(a, b), (b, c)}, the smallest equivalence relation containing R is {(a, a), (b, b), (c, c), (d, d), (a, b), (b, a), (a, c), (b, c), (c, a), (c, b)}. ]

The Attempt at a Solution

 

[LCKurtz]

Well, you know you need all pairs (a,a) for reflexivity and all pairs (2a,b) and (a,2b) for symmetry. So the question becomes if you do have all those, is that enough to get transitivity or do you need more?