Discrete Math Review?

Question

I am studying for a midterm and this question was on the review.
Let S be a relation in Z. Z being any integer. Defined as aSb if and only if a/b = 4^k k being an integer. Prove that S is symmetric and find [1]. This is equivalent.
Here are my steps but I do not know if it is right:

Proof:
a/b = 2^k b/c = 2^n = a/c = 2^k+n = a/c = 2^p

[1] = { x E Z : aSb} E = within sign not on keyboard
= { x E Z : x/1 = 2^k k E Z }
= { x E Z : x = 2^k *1 k E Z }
= { 1,4,1/4, 16, 1/16…}

Are they right? Am I missing any steps in the problems?

I was confusing symetric for transitive. I keep getting this answer is this what is supposed to be

[1] = { x E Z : aSb} E = within sign not on keyboard
= { x E Z : x/1 = 2^k k E Z }
= { x E Z : x = 2^k *1 k E Z }
= { 1,4,1/4, 16, 1/16…}

Answer 1

Puggy is right, and symmetry is very easy to prove in this case:

If a/b = 4^k,
then
b/a = 4^(-k), and since k is an integer, so is -k

i.e. if aSb then bSa

Answer 2

aSb is defined as a/b = 4^k, where k is an integer.

To prove that S is symmetric, we must prove that if aSb, then
bSa.

I think what you proved is transitivity and not symmetry.