# Partiel order

• Oct 19th 2009, 02:23 PM
Partiel order
Is there a smart way to dertermine wether a relation is called a partiel order.

For a relation to be a partiel order (dont know how you describe it) it has to be reflexive, antisymmetric and transitive.

reflexive: aRb <=> aRa
tansitive aRb <=> aRb and bRc => aRc
antisymmetric: aRb <=> aRb and bRa => a=b

Can you help me out to dertermine these?

xR1y <=>x + y = or <0

xR2y <=>x y = or <0

xR3y <=>x + y < 0
• Oct 19th 2009, 02:56 PM
Plato
Quote:

Is there a smart way to dertermine wether a relation is called a partial order.

For a relation to be a partial order (dont know how you describe it) it has to be reflexive, antisymmetric and transitive.

reflexive: aRb <=> aRa
tansitive aRb <=> aRb and bRc => aRc
antisymmetric: aRb <=> aRb and bRa => a=b

Can you help me out to dertermine these?

xR1y <=>x + y = or <0

xR2y <=>x y = or <0

xR3y <=>x + y < 0

$\displaystyle (-2,-1)\in R_1~\&~(-1,-2)\in R_1\text{ but is it true that }-2\ne -1?$

$\displaystyle (-2,-1)\in R_2\text{ but is it true that }(-1,-2)\in R_2 ?$

$\displaystyle \text{Is it true that }(0,0)\in R_3 ?$
• Oct 19th 2009, 03:10 PM
Quote:

Originally Posted by Plato
$\displaystyle (-2,-1)\in R_1~\&~(-1,-2)\in R_1\text{ but is it true that }-2\ne -1?$

here you see if the if the relation is antisymmetric? which is true if it had been "x not related to y."

$\displaystyle (-2,-1)\in R_2\text{ but is it true that }(-1,-2)\in R_2 ?$

I can't see what you are doing here.

$\displaystyle \text{Is it true that }(0,0)\in R_3 ?$

here you see if its reflexive which is isnt.

can you elaborate on how you decide wether its partiel or not.
• Oct 19th 2009, 03:25 PM
Plato
Quote:

here you see if its reflexive which is isnt.
can you elaborate on how you decide wether its partiel or not.

First the English spelling is partial.

I showed that $\displaystyle R_2$ is not symmetric.

Also showed that $\displaystyle R_1$ is not antisymmetric.
• Oct 19th 2009, 03:32 PM
Quote:

Originally Posted by Plato
First the English spelling is partial.

I showed that $\displaystyle R_2$ is not symmetric.

Also showed that $\displaystyle R_1$ is not antisymmetric.

But for a relation to be partial it is not needed to be symmetric, or?
• Oct 19th 2009, 03:48 PM
Plato
Quote:

But for a relation to be partial it is not needed to be symmetric, or?

You are correct about that. I wanted to show how things work.
Is $\displaystyle R_2$ transitive?
• Oct 19th 2009, 04:01 PM
Quote:

Originally Posted by Plato
You are correct about that. I wanted to show how things work.
Is $\displaystyle R_2$ transitive?

I can't really figure that out, Im afraid.

Maybe something with -1 - (-2) < or = 0 => -2 - z < or = 0

then -1 - z < or = 0.

Im quite lost acutally.
• Oct 19th 2009, 04:21 PM
Plato
Do you understand this whole area?
$\displaystyle \left\{ \begin{gathered} x - y \leqslant 0 \hfill \\ y - z \leqslant 0 \hfill \\ \end{gathered} \right.\, \Rightarrow \,x - z \leqslant 0$
• Oct 19th 2009, 04:29 PM
Isnt that basically what i wrote about transitivity?

xRy and yRz => xRz

However, I have difficulty seeing how yRz can be proven.
• Oct 19th 2009, 04:33 PM
Plato
Quote: