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Given any function ''f'' in ''n'' variables with values in an abelian group, a symmetric function can be constructed by summing values of ''f'' over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over [[even permutation]]s and subtracting the sum over [[odd permutation]]s. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions ''f''. The only general case where ''f'' can be recovered if both its symmetrization and anti-symmetrization are known is when ''n'' = 2 and the abelian group admits a division by 2 (inverse of doubling); then ''f'' is equal to half the sum of its symmetrization and its anti-symmetrization.
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<math>f(x_1,x_2,x_3)=(x-x_1)(x-x_2)(x-x_3)</math>
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<math> (x-x_1)(x-x_2)(x-x_3)=(x-x_2)(x-x_1)(x-x_3)=(x-x_3)(x-x_1)(x-x_2)</math>, and so on, for all permutations of <math>x_1,x_2,x_3</math>
2) Consider the circle function:
<math>f(x,y)=x^2+y^2-r^2</math>
If the x,y variables are interchanged the function becomes
<math>f(x,y)=y^2+x^2-r^2</math>
,which yields gives exactly the same results as the original f(x,y). In this case, the symmetry of the function can be seen as a symmetry of rotation of the circle around the axes x and y.
3) Consider now the ellipse equation:
<math>f(x,y)=(\frac{x}{a})^2+(\frac{y}{b})^2-r^2</math>
If x and y are interchanged, the function becomes
<math>f(x,y)=(\frac{y}{a})^2+(\frac{x}{b})^2-r^2</math>
,where we effectively swapped the two semi axes.
== Applications ==
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