On the set
$\displaystyle
\mathbb{P}\times{}\mathbb{P}
$
define a relation by (x,y)~(u,v) if and only if xv = yu.
(a) Show that ~ is an equivalence relation.
(b) Find the equivalence class of (2,3)
thanks
Im assuming we are in a commutative setting here.
Reflexive
$\displaystyle (a,b)\sim(a,b)$?
$\displaystyle ab=ba$
Symmetry
$\displaystyle (a,b)\sim (c,d)\Rightarrow ad=bc \Rightarrow cb=da \Rightarrow (c,d) \sim (a,b)$
Think you can do transitive?
(2,3), so the things equivalent are of the form $\displaystyle (a,b)$ and must satisfy $\displaystyle 2b=3a\Rightarrow b=\frac{3}{2}a$, so the equivalence class is
$\displaystyle \{(a, \frac{3}{2}a)|a\in \mathbb{P})\}$
$\displaystyle (x,y)\sim(u,v) \Leftrightarrow xv = yu$
~ is reflexive iff $\displaystyle \forall {a,b} \in {\mathbb{P}} : (a,b)\sim(a,b) \$
$\displaystyle (a,b) \sim (a,b) \Leftrightarrow ab = ba$
Since $\displaystyle \mathbb{P}$ is commutative, $\displaystyle ab=ba$ for any $\displaystyle {a,b} \in \mathbb{P}$
In other words (a, b) is in the same equivalence class as (2, 3) if and only if $\displaystyle \frac{b}{a}= \frac{3}{2}$. You could think of that class as all fractions that are equivalent to the fraction $\displaystyle \frac{3}{2}$ and so representing the rational number $\displaystyle \frac{3}{2}$.