- #1
AlonsoMcLaren
- 90
- 2
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?
If so, how to prove it?
The equation f(x,y)+f(y,x)=0 represents a special type of function called an anti-symmetric function. This means that the value of the function is equal to the opposite of its value when the inputs are switched. In other words, f(x,y)=-f(y,x).
The equation f(x,y)+f(y,x)=0 can help us determine if a function is anti-symmetric. If a function satisfies this equation for any x and y, then it is anti-symmetric. This information can be useful in studying the properties of a function and making calculations.
Yes, it is possible for a function to be equal to g(x)-g(y) for any x and y. This would be the case if g(x) is an anti-symmetric function, since f(x,y) would also be anti-symmetric. However, this is not always true. There are other types of functions that can satisfy the equation f(x,y)+f(y,x)=0, but may not be equal to g(x)-g(y).
No, it is not always true that f(x,y) is equal to g(x)-g(y) for all x and y. In order for this to be true, g(x) would have to be an anti-symmetric function and f(x,y) would also have to be anti-symmetric. However, there are other types of functions that can satisfy the equation f(x,y)+f(y,x)=0, but may not be equal to g(x)-g(y).
In order to prove that f(x,y)=g(x)-g(y), you would need to show that f(x,y) and g(x)-g(y) are equal for all possible values of x and y. This can be done by substituting different values for x and y into the equation and checking if both sides are equal. If they are, then you have proven that f(x,y) is equal to g(x)-g(y) for all x and y.