Symmetric Difference of Sets

The symmetric difference of two sets A and B is the set of all elements that belong to exactly one of the two sets. In other words, it contains the elements that are in A or in B, but not in both. $$ A \Delta B = (A \setminus B) \cup (B \setminus A) $$

The symmetric difference is a useful operation for identifying the elements that are unique to each set while excluding those the sets have in common.

It is formally defined as:

$$ A \Delta B = (A \setminus B) \cup (B \setminus A) $$

where:

• \(A \setminus B\) is the set of elements that belong to A but not to B.
• \(B \setminus A\) is the set of elements that belong to B but not to A.

Equivalently, the symmetric difference is the union of the two relative complements \(A \setminus B\) and \(B \setminus A\).

In a Venn diagram, the symmetric difference is represented by the two non-overlapping regions of the circles representing the sets.

A Practical Example

Consider the following sets:

$$ A = \{1, 2, 3\} $$

$$ B = \{2, 3, 4\} $$

First, find the elements that belong only to A and only to B:

$$ A \setminus B = \{1\} $$

$$ B \setminus A = \{4\} $$

Next, combine these two sets:

$$ A \Delta B = \{1\} \cup \{4\} = \{1, 4\} $$

Therefore, the symmetric difference of A and B is:

$$ A \Delta B = \{1, 4\} $$

The Venn diagram below highlights the elements that belong to exactly one of the two sets.

Example 2

Now consider another pair of sets:

$$ A = \{a, b, c\} $$

$$ B = \{c, d, e\} $$

Again, start by finding the elements that are unique to each set:

$$ A \setminus B = \{a, b\} $$

$$ B \setminus A = \{d, e\} $$

Then take their union:

$$ A \Delta B = \{a, b\} \cup \{d, e\} = \{a, b, d, e\} $$

So, the symmetric difference of A and B is:

$$ A \Delta B = \{a, b, d, e\} $$

The following Venn diagram illustrates the result.

Properties of the Symmetric Difference

The symmetric difference satisfies several important algebraic properties:

• Commutativity
  Changing the order of the sets does not change the result. \[ A \Delta B = B \Delta A \]
• Associativity
  The sets can be grouped in different ways without affecting the result. \[ A \Delta (B \Delta C) = (A \Delta B) \Delta C \]
• Identity Element
  The empty set acts as the identity element for the symmetric difference. \[ A \Delta \emptyset = A \]
• Self-Inverse Property
  The symmetric difference of a set with itself is the empty set. \[ A \Delta A = \emptyset \]

These properties make the symmetric difference a fundamental operation in set theory, Boolean algebra, computer science, and many other areas of mathematics.