Equivalence relations typically use the notation $\sim$. But say that I am trying to introduce a relation and prove that it is an equivalence relation. Is it incorrect to use this notation? Does it require that I already know that a relation satisfies the necessary properties? Is the same true for the notation $a \equiv b$?
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2Use a notation and define what it means. The only problem is if the notation has a "usual" meaning that is "close" to what you are defining, and this can be handled by either adjusting your notation (such as using an extra mark like a tick, umlaut, etc.) or else being extra clear each time you use your notation. – abiessu Mar 31 '21 at 03:01
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I understand, but I'm not sure what the usual notation here is, because I've seen it done in both ways. – user861776 Mar 31 '21 at 03:03
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Youn could use $\sim$, as in this very nice Brian Scott's answer. If you need more examples, just type "Equivalence relation" in the search box. – J.-E. Pin Apr 04 '21 at 05:45
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The '$\,\sim\,$' symbol is often used to denote "is related to" by whatever relation you are discussing at the time. The '$\,\equiv\,$' symbol usually denotes "is equivalent to" by whatever equivalence relation you are discussing; and that is most usually a standard form of equivalence.
You could use '$\,\equiv\,$' in this case, but more typically we reserve it for relations that are already known to be equivalences, rather than those we wish to prove are such.
Graham Kemp
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