If f(x,y)+f(y,x)=0 for any x,y, is it true that f(x,y)=g(x)-g(y)

Thread starter AlonsoMcLaren
Start date Apr 29, 2012

AI Thread Summary

The discussion centers on whether the condition f(x,y) + f(y,x) = 0 implies that f(x,y) can be expressed as g(x) - g(y). It is concluded that this is not necessarily true, as alternative forms like yg(x) - xg(y) also satisfy the original relation. The participants note that a general expression for f(x,y) can be f(x,y) = g(x,y) - g(y,x). This indicates that multiple functions can fulfill the condition, not just the proposed form. The conversation emphasizes the need for careful consideration of function definitions in such mathematical contexts.

Apr 29, 2012

AlonsoMcLaren

If f(x,y)+f(y,x)=0 for any x,y, is it true that f(x,y) can always be written as g(x)-g(y)?

If so, how to prove it?

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Apr 29, 2012

sweet springs

No. For example yg(x)-xg(y) also satisfies the relation.
Regards.

Apr 29, 2012

AlonsoMcLaren

Thanks

Apr 30, 2012

sweet springs

In general f(x,y)=g(x,y)-g(y,x) obviously.

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