Geometric interpretation of $\mathbb{R}\times\mathbb{R}/H$ for these $H$

Question

$\mathbb{R}\times\mathbb{R}/H$ 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.

Answer 1

An idea: try to find out what two equal elements modulo $\;H\;$ look like, say in the first case:

$$(a,b)+H=(x,y)+H\iff (a-x, b-y)\in H\iff b-y=0\iff b=y$$

and this means every horizontal line $\;y=k\;,\;\;k\in\Bbb R\;\;\text{a constant}\;$ , yields one single element in the quotient.

Or in the second case:

$$(a,b)+H=(x,y)+H\iff (a-x, b-y)\in H\iff b-y=x-a\iff a+b=x+y$$

so that every line of the form $\;y=-x+k\;,\;\;k\in\Bbb R\;\;\text{a constant}\;$ , yields the same element, and etc.

Observe that in both cases above we get lines that are a translation of the one defining $\;H\;$ ...

Answer 2

Geometrical meaning of quotient is "gluing" points which differ by element of $H$. If $H$ is a line, for any point $(x, y)$ you "glue" all points which are lying on line parallel to one described by $H$ and going through (x, y). Therefore an element of quotient space is not a point (x, y), but all points on line described above. And therefore quotient space can be thought as space of all lines parallel to one described in $H$.