Three question about binary relations

Question

First question. The binary relation $\geq$ on $\mathbb{R}^2$ by $(x_1, x_2) \geq (y_1, y_2)\iff x_1 \geq y_1$ and $x_2 \geq y_2$. How can I prove that $\geq$ is a preorder but not a weak order? ($\geq$ is reflexive and transitive but not complete.) I really don't understand why it is not complete?

The binary relation $\geq$ on $\mathbb{R}^2$ by $(x_1, x_2) \geq (y_1, y_2)\iff x_1 > y_1 $ or $x_1=y_1$ and $x_2>y_2$. How can I show that $\geq$ is a linear order?

Second question. If $P$ is asymmetric and negatively transitive, then it is irreflexive, transitive and acyclic. How can I prove that?

Third question. $P$ and $R$ are relations. $xRy \iff (y,x) \notin P$. How can I show that $P$ is asymmetric iff $R$ is complete and that $P$ is negatively transitive iff $R$ is transitive?

Answer 1

The relation $\geq$ in your first question is not complete, since you can give a pair where you can not dicide if $(x_1, x_2)\geq (y_1, y_2)$. Take for example $(1,3)$ and $(0,4)$.

At least that is what I understand under 'complete', since I am not familiar with the phrase in that context. Consider to give a definition.

Then we have neither $(1,3)\geq (0,4)$ nor $(0,4)\geq (1,3)$.

It is $1\geq 0$ but we do not have $3\geq 4$. And it is $4\geq 3$ but not $0\geq 1$.

The binary relation $\geq$ on $\mathbb{R}^2$ by $(x_1, x_2) \geq (y_1, y_2)\iff x_1 > y_1 $ or $x_1=y_1$ and $x_2>y_2$. How can I show that $\geq$ is a linear order.

This is the so called 'lexicographical order'. To show that this is a linear order, you have to show that it satisfies the definition.

So it is reflexiv, antisymmetric, transitiv and totally(?) (or complete(?))

Just write down what there is to show. The proofs are easy.