Non-Symmetric Function

A Non-Symmetric Function is a function that does not remain unchanged under any permutation of its input variables.

• Context:
  • It can violate Permutation Symmetry by yielding different outputs when the order of its Input Variables is changed.
  • It can be linked to Asymmetric Relations or Order-Dependent Functions, where changes in the order of inputs directly affect the result.
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• Example(s):
  • The function [math]\displaystyle{ f(x, y) = x - y }[/math], where [math]\displaystyle{ f(x, y) \neq f(y, x) }[/math].
  • The Kullback-Leibler (KL) Divergence [math]\displaystyle{ D_{KL}(P \parallel Q) \neq D_{KL}(Q \parallel P) }[/math], which measures the difference between two probability distributions but is not symmetric in its arguments.
  • The function [math]\displaystyle{ f(x, y) = \frac{x}{y} }[/math], which is not symmetric since [math]\displaystyle{ \frac{x}{y} \neq \frac{y}{x} }[/math].
  • The cross product of two vectors in 3D space, [math]\displaystyle{ \mathbf{a} \times \mathbf{b} \neq \mathbf{b} \times \mathbf{a} }[/math], which is a classic non-symmetric operation in vector algebra.
  • The function [math]\displaystyle{ f(x, y) = \max(x, y) - \min(x, y) }[/math], which changes when [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are swapped.
  • ...
• Counter-Example(s):
  • A Symmetric Function like [math]\displaystyle{ f(x, y) = x + y }[/math], where [math]\displaystyle{ f(x, y) = f(y, x) }[/math].
  • A Distance Metric Function satisfying symmetry, such as the Euclidean distance, where [math]\displaystyle{ d(x, y) = d(y, x) }[/math] for any points [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math].
  • A Symmetric Polynomial such as [math]\displaystyle{ f(x, y, z) = x y + y z + z x }[/math], which remains unchanged under permutations of its variables.
• See: Symmetric Function, Anti-Symmetric Function, Permutation, Asymmetric Relation, Order-Dependent Function, Non-Symmetric Relation, Non-Symmetric Similarity Measure.

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