Symmetry
Symmetry
A dyadic relation R is symmetric if, for all x and y in the field of R, xRy ? yRx; it is asymmetric if, for all x and y in the field of R, xRy ? ~ yRx; non-symmetric if there are x and y in the field of R such that [xRy] [~ yRx]. An n-adic propositional function F is symmetric if F(x1, x2, . . . , xn) is materially equivalent to the proposition obtained from it by permuting x1, x2, . . . , xn among themselves in any fashion — for all sets of n arguments x1, x2, . . . , xn belonging; to the range of F.
A dyadic function f, other than a propositional function, is symmetric if, for all pairs of arguments, x, y, belonging to the range of f, f(x, y) = f(y, x). An n-adic function f is symmetric if, for any set of n arguments belonging to the range of f, the same value of the function is obtained no rmtter how the arguments are permuted among themselves (i.e., if the value of the function is independent of the order of the arguments).
In geometry, a figure is said to be symmetric with respect to a point P if the points of the figure can be grouped in pairs in such a way that the straight-line segment joining any pair has P as its midpoint. A figure is symmetric with respect to a straight line l if the points can be grouped in pairs in such way that the straight-line segment joining any pair has l as a perpendicular bisector. These definitions apply in geometry of any number of dimensions. Similar definitions may be given of symmetry with respect to a plane, etc. — A.C.