TITLE: Equivalence Relation on $(\mathbb{R} - \{0\}) \times \mathbb{R}$ defined by $(x, y) \sim (u, v) \iff xv=yu$
QUESTION [1 upvotes]: Looking for some help / direction as equivalence relations are new to me.
Let $U$ be the equivalence relation on $(\mathbb{R} - \{0\}) \times \mathbb{R}$ defined by $(x, y)$ is $U$-related to $(u, v)$ if and only if $xv=yu$.
a.)  Write the equivalence class $[(2,3)]_U$ using set notation.  Geometrically, what is $[(2, 3)]_U$?
b.)  Write the partition of $(\mathbb{R} - \{0\}) \times \mathbb{R}$  corresponding to the relation $U$.

REPLY [2 votes]: Assuming you meant $[(2,3)]_U$, the equivalence class of $(2,3)$ with respect to the equivalence relation $U$, let's look at what that looks like.
$$(u,v) \sim (2,3) \iff 2v = 3u \iff v = \frac{3}{2}u$$
This s a line in the plane going through the origin with slope $3/2$. In set builder notation:
$$[(2,3)]_U = \{(u,v) \in (\Bbb R \setminus \{0\})\times\Bbb R : 2v = 3u\}$$

One way to partition the space is 
$$\bigl\{[(1,m)]_U : m \in \Bbb R\bigr\}.$$
The equivalence class $[(1,m)]_U$ corresponds to a line through the origin with slope $m$. The lines are distinct for distinct $m$ and cover the whole space. (Note that a vertical line has undefined slope, but we don't worry about this since $\{0\}$ has been removed from the space.)
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