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I am having a difficult time trying to give words to an equivalence, namely, $\preceq$; for instance, how would I word $x \preceq y$? I know it's not that x is less than or equal to y, because $\preceq$ is just a generalization of different equivalences.

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I would not expect to see the symbol $\preceq$ used for an equivalence relation; rather, I’d expect it to be used for a partial order (or one of its generalizations that you probably haven’t encountered yet). –  Brian M. Scott Nov 16 '12 at 0:21
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If $\preceq$ is some kind of order (partial or total) you can read $$x \preceq y$$

as

  • $x$ precedes or is equal to $y$
  • $x$ is less than or equal to $y$
  • $x$ is smaller or equal to $y$

depending on context. Else, you might just read $x$ is in relation with $y$. Note that the code for this is \preceq which is an abbreviation of "precedes" and "equal".

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