If $f(x,y) = f(y,x)$, does it follow that $x=y$?
If yes, please show a proof. If no, please demonstrate a counter-example.
Thank you.
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closed as not constructive by Andres Caicedo, dragoncharmer, Paul, Did, Thomas Oct 25 '12 at 13:00
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or specific expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, see the FAQ for guidance.
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No. It is false. For instance, $$f(x,y) = \vert x - y \vert^k$$ $$f(x,y) = g(x) g(y)$$ $$f(x,y) = h(x) + h(y)$$ |
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It's totally false. Take any constant function defined on $X \times X$. |
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There are numerous counterexamples, such as $f(x,y)=xy$. In fact, any symmetry function of satisfies that $f(x,y)=f(y,x)$ for every $(x,y)$ in the domain. |
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