Symmetric Functions Math Example 1

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

Example 1

easy
Is the function f(x,y)=x2+y2+xyf(x, y) = x^2 + y^2 + xy symmetric in xx and yy?

Solution

  1. 1
    A function is symmetric in xx and yy if swapping xx and yy gives the same function: f(x,y)=f(y,x)f(x,y) = f(y,x).
  2. 2
    Compute f(y,x)=y2+x2+yx=x2+y2+xyf(y,x) = y^2 + x^2 + yx = x^2 + y^2 + xy.
  3. 3
    Since f(x,y)=f(y,x)f(x,y) = f(y,x), the function is symmetric in xx and yy.

Answer

Yes, f(x,y) is symmetric in x and y\text{Yes, } f(x,y) \text{ is symmetric in } x \text{ and } y
A symmetric function remains unchanged when its variables are interchanged. The elementary symmetric polynomials (like x+yx+y, xyxy, x2+y2x^2+y^2) are building blocks for all symmetric functions, by the fundamental theorem of symmetric polynomials.

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 2 medium

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

Example 3 medium

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

Example 4 hard

Determine whether [formula] is a symmetric function of three variables.

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