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I have a relation $\mathbb{R\subseteq M\times M}$, which is not even a partially order :

$$ M=\{x\in\mathbb{R}:-3\leq x\leq3\}, R=\{(x,y):x>y\}. $$

Well what I'm trying to do is to make this relation a total order (linear order). And for this operation I can add or erase some elements as little as possible. Finally I have to give the $\min, \max, \sup, \inf$ of the set $K\subseteq M$.

Here is the set of $K$:


I hope you can help me!

Thanks in advance!

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Can you please clean up your question? For example, what are the sets $\mathbb R$ and $\mathbb M$? Are they the same as $R$ and $M$? Also, is this homework? If so, please add the homework tag. – Alex Becker Jan 5 '13 at 4:30
Of course , i think i can explain. Well the relation is with "R" defined with a condition "x greater than y". "M" is the set of the relation. And "K" is the poset of "M". – Tiro Jan 5 '13 at 4:37
No this is not a homework. This is just an exercise from a older exam of my school. And i need to solve an understand this for the preparation of my real exam.Thank you! – Tiro Jan 5 '13 at 4:40
I get what $R$ and $M$ are. But what are $\mathbb R$ and $\mathbb M$ in the first line? You never define the blackboard versions. While I can assume $\mathbb R$ is the real numbers, I suspect this is not what you mean. – Alex Becker Jan 5 '13 at 4:42
$\mathbb{R\subseteq M\times M}$ means that : A binary relation R is a subset of the Cartesian product of two sets A and B: – Tiro Jan 5 '13 at 4:48

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