Partiel order

**Madspeter** · October 19th 2009, 02:23 PM

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

**Plato** · October 19th 2009, 02:56 PM

Originally Posted by Madspeter

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

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

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

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

**Madspeter** · October 19th 2009, 03:10 PM

Originally Posted by Plato

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

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

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

I can't see what you are doing here.

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

here you see if its reflexive which is isnt.

can you elaborate on how you decide wether its partiel or not.

**Plato** · October 19th 2009, 03:25 PM

Originally Posted by Madspeter

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 $R_2$ is not symmetric.

Also showed that $R_1$ is not antisymmetric.

**Madspeter** · October 19th 2009, 03:32 PM

Originally Posted by Plato

First the English spelling is partial.

I showed that $R_2$ is not symmetric.

Also showed that $R_1$ is not antisymmetric.

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

**Plato** · October 19th 2009, 03:48 PM

Originally Posted by Madspeter

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 $R_2$ transitive?

**Madspeter** · October 19th 2009, 04:01 PM

Originally Posted by Plato

You are correct about that. I wanted to show how things work.
Is $R_2$ 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.

**Plato** · October 19th 2009, 04:21 PM

Do you understand this whole area?
$\left\{ \begin{gathered}<br /> x - y \leqslant 0 \hfill \\<br /> y - z \leqslant 0 \hfill \\ <br /> \end{gathered} \right.\, \Rightarrow \,x - z \leqslant 0$

**Madspeter** · October 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.

**Plato** · October 19th 2009, 04:33 PM

Originally Posted by Madspeter

Isnt that basically what i wrote about transitivity?
xRy and yRz => xRz
However, I have difficulty seeing how yRz can be proven.

You are in need of some 'sit down' help.
Please go to your instructor/tutor for more help.

**Madspeter** · October 19th 2009, 04:37 PM

Im afraid I can't afford that.

Thanks for your help anyway.

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