$\mathbb{R}\times\mathbb{R}/H$ with: ($G=\mathbb{R}\times\mathbb{R}$)
$H=\{(x,0)|x\in\mathbb{R}\}$
$H=\{(x,y)|y=-x\}$
$H=\{(x,y)|y=2x\}$
This is from a book (Pinter's "A book of abstract algebra") in a chapter before use of the isomorphism theorems (so Understanding the quotient of infinite groups $\mathbb{R}^2/H$ where $H = \{(a, 0): a\in \mathbb{R}\}$ doesn't apply)
The question asks: "In geometric terms describe the elements of G/H" and "in geometric terms describe the operation of G/H"
I'm quite stuck. For example with the first one, $\mathbb{R}\times\mathbb{R}/H=\{(x+u,v)|x,u,v\in\mathbb{R}\}$
Which doesn't really help.
There are no answers for this in the back of the book, so I'm not sure what I'm even looking for.
Addendum
Slight mistake with the first one, $G/H=\{\{(x+u,v)|x\in\mathbb{R}\}|u,v\in\mathbb{R}\}$ is what it should read, which is quite simply "the set of all horizontal lines"
I will try to proceed (this is my third error) - I would still like to know what I should be spotting regarding the geometric interpretation.