1. ## 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

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

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

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

3. 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.

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.

5. 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?

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?

7. 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.

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

9. Isnt that basically what i wrote about transitivity?

xRy and yRz => xRz

However, I have difficulty seeing how yRz can be proven.