Symmetric Difference

I was recently flipping through a real analysis textbook, and came across a notation that I had not seen before. First, the set difference of two sets \(A\) and \(B\) (which I had seen before) was defined as \[ A \setminus B = A \cap B^c. \] Here \(B^c\) is the set compliment of \(B\), or all elements not in \(B\). You can equivalently write the set difference as \[ A \setminus B = \{x \, |\, x \in A\, \text{and}\, x \notin B\} \]

Then a new set operation was defined as follows \[ A \triangle B = (A \setminus B) \cup (B \setminus A). \] This operation was not given a name, but I looked it up and discovered that \(A \triangle B\) is called the “symmetric difference” of the sets \(A\) and \(B\). In words, it is the union of both set differences. An equivalent, but more intuitive, way to express the symmetric difference is \[ A \triangle B = (A \cup B) \setminus (A \cap B), \] which is the union minus the intersection.

This operation can be implemented in Mathematica (I got this implementation from Define

SymmetricDifference[a_, b_] :=
    Union[Complement[a, b], Complement[b, a]]

For example, if I define the sets

a = {1, 2, 3, 4, 5};
b = {4, 5, 6, 7, 8};

then SymmetricDifference[a,b] returns {1,2,3,6,7,8}.

The notation \(A \triangle B\) does risk confusion with the Laplace operator on a function \(f\): \[ \Delta f = \nabla \cdot \nabla f. \] In LaTeX, I used \triangle for the symmetric difference, and \Delta for the Laplacian. The MathWorld article linked above recommends using \ominus, so that the symmetric difference of sets \(A\) and \(B\) is given by \[ A \ominus B. \]

Landon Lehman
Data Scientist

My research interests include data science, statistics, physics, and applied math.