Cosets and Equivalence Relations Today in math my professor was talking about equivalence relations and modulo. Then he said something about cosets, and I don't understand the relation. I read that a coset was a subgroup with all if it's elements added with another element of the group. And I guess equivalence relations for subgroups, at least the integers mod some number?
My main question is how are cosets and equivalence relations related?

Cosets and equivalences relations are not related, except inasmuch as you can define an equivalence relation fairly naturally on the cosets of some subgroup of a group.
For example, taking the group $G = (\mathbb{Z}, +)$, and its subgroup $(3\mathbb{Z},+)$ gives you an infinite number of cosets $(\dots,-1 + 3\mathbb{Z}, 0 + 3\mathbb{Z}, 1 + 3\mathbb{Z}, 2 + 3\mathbb{Z}, 4 + 3\mathbb{Z},\dots)$. However, as $1 + 3\mathbb{Z}$ and $4 + 3\mathbb{Z}$ define the same sets, they are equivalent, both being the set of numbers equivalent to $1$ mod $3$.
