Symmetric Functions Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Determine whether f(x,y,z)=x2y+y2z+z2xf(x, y, z) = x^2y + y^2z + z^2x is a symmetric function of three variables.

Solution

  1. 1
    Test by swapping xx and yy: f(y,x,z)=y2x+x2z+z2yf(y, x, z) = y^2x + x^2z + z^2y. Compare with f(x,y,z)=x2y+y2z+z2xf(x,y,z) = x^2y + y^2z + z^2x.
  2. 2
    These are NOT equal (e.g., x=1,y=2,z=0x=1, y=2, z=0: f(1,2,0)=2f(1,2,0)=2, f(2,1,0)=0+0+0=0f(2,1,0)=0+0+0=0). So ff is NOT symmetric. However, ff is a cyclic function: f(y,z,x)=y2z+z2x+x2y=f(x,y,z)f(y,z,x) = y^2z + z^2x + x^2y = f(x,y,z).

Answer

Not symmetric, but cyclic\text{Not symmetric, but cyclic}
A function is symmetric if it is invariant under ALL permutations of its variables. A function is cyclic if it is invariant under cyclic permutations (x→y→z→xx \to y \to z \to x) but not necessarily transpositions. Cyclic functions are a broader class that includes symmetric functions as a special case.

About Symmetric Functions

A symmetric function is one that remains unchanged (or changes in a predictable way) under specific variable transformations. Even functions satisfy f(−x)=f(x)f(-x) = f(x) and are mirror-symmetric about the y-axis; odd functions satisfy f(−x)=−f(x)f(-x) = -f(x) and have 180-degree rotational symmetry about the origin.

Learn more about Symmetric Functions →

More Symmetric Functions Examples

Example 1 easy

Is the function [formula] symmetric in [formula] and [formula]?

Example 2 medium

Express [formula] in terms of the elementary symmetric polynomials [formula] and [formula].

Example 3 medium

If [formula] and [formula], find [formula].

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