# If $f(x,y) = f(y,x)$, does it follow that $x=y$? [closed]

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, hollow7, Paul, Did, ThomasOct 25 '12 at 13:00

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It probably would have been good to show your work on this one. The fact that you didn't and the huge number of counterexamples makes people assume you didn't even attempt the problem, which is where I think the downvotes are coming from. –  Robert Mastragostino Oct 25 '12 at 5:33
Yes, I must admit to being confused and not thinking through the problem carefully. It's just that I recently saw many problems where f(x,y) = f(y,x) and the solution is just to solve for f(y,y). –  hollow7 Oct 25 '12 at 5:36
I think any continuous function $\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is a counter-example. –  Douglas S. Stones Oct 25 '12 at 5:54
I must admit to .../... not thinking through the problem carefully... Why is that that you did not? –  Did Oct 25 '12 at 6:35

No. It is false. For instance, $$f(x,y) = \vert x - y \vert^k$$ $$f(x,y) = g(x) g(y)$$ $$f(x,y) = h(x) + h(y)$$
It's totally false. Take any constant function defined on $X \times X$.
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.