I'm currently doing some work on GT from a book, and I'm stuck on the following questions. Can anyone please help?

1. "Let G be a group with a binary operation written multiplicatively, and H be a subgroup of G.

a) Shouw that the relation ~ defined by: x ~ y if there are

,

(element) H such that x

= y

is an equivalence relation.

b) For x (element of) G, show that [x], the equivalence class of x is equal to the coset xH"

2. "Let G be a group with a binary operation written multiplicatively. Suppose H is a non-empty subset of G such that for all x,y (element of) H, we have

(element of) H. Prove that H is a subgroup of G."

These are the last 2 questions in the chapter and I honestly have no clue how to do them