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)

(Itwasntme)

thanks

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- Aug 19th 2009, 07:21 PMquah13579Discrete set maths
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)

(Itwasntme)

thanks - Aug 19th 2009, 07:45 PMGammanot sure what your sets P are, but...
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

$\displaystyle \{(a, \frac{3}{2}a)|a\in \mathbb{P})\}$ - Aug 24th 2009, 11:36 PMyoonsi
I'm sorry but could you explain that again? I'm confused, ab = ba, how does that show that its reflexive? and I thought that (x,y) =/= (y,x)

Thank you =] - Aug 25th 2009, 12:35 AMquah13579
- Aug 25th 2009, 02:09 AMDefunkt
$\displaystyle (x,y)\sim(u,v) \Leftrightarrow xv = yu$

~ 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 $\displaystyle {a,b} \in \mathbb{P}$ - Aug 26th 2009, 11:17 PMyoonsi
oh well I understand the concept of reflexivity buuuut, how come P is commutative? I thought in ordered pairs, (2,3) does not equal (3,2)?

Sorry I must seem really silly...

Could someone also step me through showing how that is transitive? (Bow) - Aug 27th 2009, 12:40 AMHallsofIvy
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**$\displaystyle \frac{3}{2}$.