For a, b belong to R\{0} define a ~b if and only if a/b belong to Q

a) prove that ~ is an equivalence relation.
b) find the equivalence class of 1.

Can anybody help me on this. thanks in advance

a) For a,b,c in R \ {0}:

Reflexive: a ~ a, because a/a = 1, which is in Q.

Transitive: Suppose that a ~ b.
So, a/b is in Q
==> b/a = 1/(a/b) is also in Q
==> b ~ a.

Symmetry: Suppose that a ~ b and b ~ c.
Then, a/b and b/c are in Q
==> a/c = (a/b)(b/c) is also in Q by closure of multiplication in Q
==> a ~ c.
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b) For the equivalence class of Q, we want to find all elements x in R \ {0} such that x ~ 1.
==> We need x/1 = x to be in Q.
Hence, the equivalence class of 1 is Q \ {0}.

I hope this helps!

Symmetry and Transitivity labels are switched; sorry!

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