The equation:

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

is a well-known set identity in set theory. It essentially states that the difference between the union and intersection of two sets is equal to the union of their symmetric differences. Let's break it down:

1. Left-hand side (LHS):

   • $A \cup B$ represents the union of sets $A$ and $B$, which includes all elements in either $A$ or $B$.
   • $A \cap B$ represents the intersection of $A$ and $B$, which includes only elements common to both $A$ and $B$.
   • Subtracting $A \cap B$ from $A \cup B$ means we are removing the common elements (the intersection) from the union.

2. Right-hand side (RHS):

   • $A - B$ represents the set of elements in $A$ but not in $B$.
   • $B - A$ represents the set of elements in $B$ but not in $A$.
   • Taking the union of $A - B$ and $B - A$ gives all elements that are in either $A$ or $B$, but not in both—this is the symmetric difference of sets $A$ and $B$.

This identity expresses that the elements belonging to exactly one of the sets $A$ or $B$ (but not both) are the same, regardless of how you compute them—whether you first compute the union and subtract the intersection, or compute the relative complements and then take their union.

Want to go over this proof or need more details?

Here are 5 related questions:

1. What is the symmetric difference of two sets?
2. Can you provide examples of union and intersection operations on sets?
3. How does this identity apply to Venn diagrams?
4. What other set identities are commonly used in set theory?
5. How does this equation apply to probability theory?

Tip: Set operations follow specific rules similar to arithmetic. Learning these rules can help in understanding more advanced topics like probability and logic!