Algebraic Symmetry Math Example 2

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

Example 2

medium
Is f(x,y)=x2−xy+y2f(x, y) = x^2 - xy + y^2 symmetric?

Solution

  1. 1
    Step 1: f(y,x)=y2−yx+x2=x2−xy+y2f(y, x) = y^2 - yx + x^2 = x^2 - xy + y^2.
  2. 2
    Step 2: f(y,x)=f(x,y)f(y, x) = f(x, y).
  3. 3
    Step 3: Yes, symmetric. Note: xy=yxxy = yx makes the middle term invariant.

Answer

Yes, symmetric.
Even expressions with cross terms (xyxy) can be symmetric, because multiplication is commutative: xy=yxxy = yx. Check by literally substituting yy for xx and xx for yy everywhere.

About Algebraic Symmetry

The property of an expression or equation that remains unchanged when certain transformations — such as swapping variables — are applied.

Learn more about Algebraic Symmetry →

More Algebraic Symmetry Examples

Example 1 easy

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

Example 3 easy

Is [formula] symmetric?

Example 4 medium

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

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