Define a relation ~ on N whereby a~b if and only if there exists such that .

(a) Prove that ~ is an equivalence relation.

(b) Write out the equivalence classes for ~ on the set {1,2,3,....10}

Here's my attempt at part (a):

To show that ~ is an equivalence relation, I must show that it's reflexive,symmetric and transitive:

* ~ is reflexive a~a

But we know: , well I'm sort of stuck here and I don't know how to conclude that ~ is reflexive...

* ~ is symmetric a~b b~a

We can suppose and then try to show that but I don't know how to do it...

* ~ is transitive if and only if a~b b~c a~c

Suppose then:

I'm not sure how to continue this part either... any help with this problem is greatly appreciated.