I am studying these concepts of order for the first time, and I am having a certain difficulty:
I define an Order relation in $A=\mathbb{R_{+}^{2}}$ as : $x,y \in A$, $x\geq y \iff x_{1} \geq y_{1}$ and $x_{2} \geq y_{2}$
Its well known that this relation is partially ordered: it is reflexive, antisymmetric, and transitive. But is not totally ordered.
Here is where begins my doubts: I dont understand why it is transitive and reflexive?
I belive if I choose a counter-example, I can show that it is not transitive and relexive, right?:
If I choose $x=(1,3)$ and $y=(\frac{1}{2},4)$ and $y=(\frac{1}{3},5)$ :
Does it reflexive? Yes. Because $x \geq x$, because $x_{1} \geq x_{1}$ and $x_{2} \geq x_{2}$
Does it transitive? How can I show that, with x, y and z, transitive condition holds? Also for the Antisymmetric condition?
What do I missing? Any help guys?
Many thanks
For transitivity, suppose $x\le y$ and $y\le z$. By definition, that means that $x_1\le y_1$, $x_2\le y_2$ and $y_1\le z_1$ and $y_2\le z_2$. Therefore, by transitivity of the order in $\mathbb{R}$, you have $x_1\le z_1$ and $x_2\le z_2$. But that means, by definition, that $x\le z$.
For antisymmetry, if you have $x\le y$ and $y\le x$ that means $x_1\le y_1\le x_1$ and $x_2\le y_2\le x_2$, which imply (by antisymmetry of the order in $\mathbb{R}$) $x_1=y_1$ and $x_2=y_2$, that is, $x=y$.