What Is Symmetric Difference?

Venn diagram
A Venn diagram with the symmetric difference of A and B shaded. C.K.Taylor

Set theory uses a number of different operations to construct new sets from old ones. There are a variety of ways to select certain elements from given sets while excluding others. The result is typically a set that differs from the original ones. It is important to have well-defined ways to construct these new sets, and examples of these include the union, intersection and difference of two sets. A set operation that is perhaps less well-known is called the symmetric difference.

Symmetric Difference Definition

To understand the definition of the symmetric difference, we must first understand the word 'or.' Although small, the word 'or' has two different uses in the English language. It can be exclusive or inclusive (and it was just used exclusively in this sentence). If we are told that we may choose from A or B, and the sense is exclusive, then we may only have one of the two options. If the sense is inclusive, then we may have A, we may have B, or we may have both A and B.

Typically the context guides us when we run up against the word or and we don’t even need to think about which way it’s being used. If we are asked if we would like cream or sugar in our coffee, it’s clearly implied that we may have both of these. In mathematics, we want to eliminate ambiguity. So the word 'or' in mathematics has the inclusive sense.

The word 'or' is thus employed in the inclusive sense in the definition of union. The union of the sets A and B is the set of elements in either A or B (including those elements that are in both sets). But it becomes worthwhile to have a set operation that constructs the set containing elements in A or B, where 'or' is used in the exclusive sense. This is what we call the symmetric difference. The symmetric difference of the sets A and B are those elements in A or B, but not in both A and B. While notation varies for the symmetric difference, we will write this as A ∆ B

For an example of the symmetric difference, we will consider the sets A = {1,2,3,4,5} and B = {2,4,6}. The symmetric difference of these sets is {1,3,5,6}.

In Terms of Other Set Operations

Other set operations can be used to define the symmetric difference. From the above definition, it is clear that we may express the symmetric difference of A and B as the difference of the union of A and B and the intersection of A and B. In symbols we write: A ∆ B = (A ∪ B) – (A ∩ B).

An equivalent expression, using some different set operations, helps to explain the name symmetric difference. Rather than use the above formulation, we may write the symmetric difference as follows: (A – B ) ∪ (B – A). Here we see again that the symmetric difference is the set of elements in A but not B, or in B but not A. Thus we have excluded those elements in the intersection of A and B. It is possible to prove mathematically that these two formulas are equivalent and refer to the same set.​

The Name Symmetric Difference

The name symmetric difference suggests a connection with the difference of two sets. This set difference is evident in both formulas above. In each of them, a difference of two sets was computed. What sets the symmetric difference apart from the difference is its symmetry. By construction, the roles of A and B can be changed. This is not true for the difference of two sets.

To stress this point, with just a little work we will see the symmetry of the symmetric difference. Since we see A ∆ B = (A – B ) ∪ (B – A) = (B – A) ∪ (A – B ) = B ∆ A.