A related concept is that of the antisymmetric tensor or alternating form. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. The space of symmetric tensors of rank r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Let f( x) be a real-valued function of a real variable, then f is even if the following equation holds for all x and -x in the domain of f:į ( x ) = f ( − x ) Main article: Even and odd functions Even functions ƒ( x) = x 2 is an example of an even function. In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (i.e., an isometry). This can occur in many ways for example, if X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. Symmetry occurs not only in geometry, but also in other branches of mathematics. The root system of the exceptional Lie group E 8. For other uses, see Symmetry (disambiguation) and Bilateral (disambiguation).
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