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Let $K$ be a proper cone. I need to prove following properties:

  1. if $x \preceq_K y$ and $u \preceq_K v$, then $x+u \preceq_K y+v$
  2. if $x \preceq_K y$ and $y \preceq_K z$, then $x \preceq_K z$
  3. if $x \preceq_K y$ and $\alpha \geq 0$, then $\alpha x \preceq_K \alpha y$
  4. $x \preceq_K x$

I can comprehend all of these in a graphical way as elements being ordered component-wise. But that does not qualify as a proof. What is the proper way of proving such properties?

zunder
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  • What do you mean by the symbol with subscript $K$? – Chee Han Jun 14 '15 at 11:39
  • That is a partial ordering induced by the proper convex cone, which is defining generalized inequalities on $\mathbb{R}^n$ – zunder Jun 14 '15 at 11:43
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    I might be wrong, but it seems like these four properties follow just by the definition of a cone. For example, if $x-y\in K$ and $y-z\in K$, then $x-y+y-z=x-z\in K$. – Chee Han Jun 14 '15 at 11:48
  • Yes, I think you are correct. Due to $x \preceq_K y \Leftrightarrow x -y\in K$ and $\theta_1\cdot x + \theta_2 \cdot y \in K$ we may write $x-y+y-z=x-z \in K \Leftrightarrow x \preceq_K z$. If you post your description as an answer, I would not mind accepting it as the right answer. – zunder Jun 14 '15 at 15:21
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    Nahh it's fine, glad I can help (: – Chee Han Jun 14 '15 at 15:41
  • @CheeHan, regarding the notation: http://math.stackexchange.com/questions/669085/what-does-curly-curved-less-than-sign-succcurlyeq-mean/669115#669115 – Michael Grant Jun 14 '15 at 20:06
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    Number 4 is particularly trivial, since it is equivalent to $0\in\mathcal{K}$. – Michael Grant Jun 15 '15 at 01:32

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