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Determine whether the relations $R$ on the real numbers $x, y$ given below are symmetric or partially ordered. Are they totally ordered?

$x \geq y$

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So, what are you having trouble with? It's just a matter of checking the definitions. –  wildildildlife Apr 17 '11 at 14:47
    
@wildildildlife : I think it's symmetric, partial ordering. I'm not sure whether it is correct. for symmetric, it is true only when x = y. –  Xingdong Apr 17 '11 at 16:35

2 Answers 2

up vote 1 down vote accepted

Pay attention to the quantifiers in the definition. Symmetry means: for all $x,y$ the implication $x\leq y\Rightarrow y\leq x$ holds.

You can't say (as you do in the comments) that the relation is symmetric for certain elements; a relation is either symmetric or not.

If you think it is symmetric, try to prove this. If not, you should find a counter-example. The latter means you should find certain elements $x,y\in\mathbb{R}$ such that $x\leq y$ holds, but $y\leq x$ does not hold.

Edit: There is only one relation: $R=\{(x,y)\in\mathbb{R}\times \mathbb{R}: x\leq y\}$. You seem to be under the impression that elements of $R$ are (also) called relations. This is not true. To say that $(x,y)\in R$ means that $x$ is related to $y$, which in the current case means that $x$ is not larger than $y$. Stated this way, symmetry of $R$ means that the implication $(x,y)\in R\Rightarrow (y,x)\in R$ holds for all pairs $(x,y)\in\mathbb{R}\times \mathbb{R}$. Or: is true that whenever $x$ is not larger than $y$, then it follows that also $y$ cannot be larger than $x$? Obviously, if $x$ and $y$ happen to be equal, then this holds. But what if $x\neq y$?

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Because the problem description doesn't say anything about the certain relation, it means all possible relations are involved or something? because the relation is one subset of cartesian product, and since it is for all real numbers , so there're infinitely number of relations, some are symmetric, some are not. So in this situation, how could I handle it ? –  Xingdong Apr 17 '11 at 18:26
    
@Xingdong: I have tried to give a better explanation in an edit. –  wildildildlife Apr 17 '11 at 19:25

Hint:

Use the definitions of symmetric, partially ordered and totally ordered to say whether the real numbers and $\ge$ meet the definition or there is a counter example.

For example, for a symmetric relation, you would need $x \ge y$ if and only if $y \ge x$. You should be able to find a counterexample.

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