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So when we talk about order relations for the familiar number systems, we are always introduced to the antisymmetry property which is $x \le y, x \ge y \implies x=y$.

When I think of the word 'antisymmetry', I think of something being the opposite of symmetry but not asymmetry. Is there any meaningful way to interpret it?

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    Maybe you can think of it this way (which I just made up). The antisymmetry of a relation $R$ is equivalent to the condition that for any two $x\ne y$, if $x R y$ then not $y R x$. So symmetry between $x R y$ and $y R x$ is forbidden for $x\ne y$.2012-08-16
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    I like that. Thanks2012-08-16
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    I've added tags [tag:relations], [tag:elementary-set-theory] and [tag:order-theory] - to me these tags seem suitable for questions about partial orders and similar relations. Feel free to change them, if you think a different set of tags would be more appropriate.2012-08-16
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    @RahulNarain: Good explanation. The $\le$ fuzzes things up a bit. But works beautifully as an explanation for $\lt$, which is where the intuition comes from anyway.2012-08-16

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