The expression $(A - B) \cup (B - A)$ represents the symmetric difference between two sets $A$ and $B$.

Definition of the Terms:

• $A - B$: The difference between sets $A$ and $B$ is the set of elements that are in $A$ but not in $B$.
• $B - A$: The difference between sets $B$ and $A$ is the set of elements that are in $B$ but not in $A$.
• $\cup$: The union of two sets includes all elements that are in either set or in both sets.

Symmetric Difference

The symmetric difference, denoted as $A \Delta B$, can be defined as:

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

Explanation

The symmetric difference between two sets $A$ and $B$ is the set of elements that are in either $A$ or $B$, but not in both. It excludes the elements that $A$ and $B$ have in common.

Example:

If:

• $A = \{1, 2, 3\}$
• $B = \{3, 4, 5\}$

Then:

• $A - B = \{1, 2\}$ (elements in $A$ but not in $B$)
• $B - A = \{4, 5\}$ (elements in $B$ but not in $A$)

So: $A \Delta B = (A - B) \cup (B - A) = \{1, 2\} \cup \{4, 5\} = \{1, 2, 4, 5\}$

Summary:

$(A - B) \cup (B - A) = A \Delta B$ This set contains all elements that are in either $A$ or $B$ but not in both.

Do you want details or have any questions?

Here are some related questions you might be interested in:

1. What is the symmetric difference if $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$?
2. How would the symmetric difference change if $A = B$?
3. Can the symmetric difference be an empty set? If so, when?
4. What is the relationship between the symmetric difference and the union of two sets?
5. How does the symmetric difference behave with more than two sets?
6. What is the complement of the symmetric difference in a universal set $U$?
7. How does the symmetric difference relate to the concept of XOR in logic?
8. How do you visualize the symmetric difference on a Venn diagram?

Tip: The symmetric difference is useful in problems where you need to identify elements that are exclusive to either of the sets. It’s like finding the elements that make two sets different from each other.