Proof of Set Difference Intersection

Prove that $(A - B) \cap (B - A) = \emptyset$.

In this problem, we are tasked with proving that the intersection of two set differences $(A - B)$ and $(B - A)$ results in an empty set. The concept of set difference plays a pivotal role in understanding the more generalized operations within set theory, particularly when dissecting complex relationships between multiple sets. Set difference essentially removes elements of one set from another, highlighting those elements that are unique to each respective set.

Given this context, when we compute the set differences $(A - B)$ and $(B - A)$, we specifically focus on elements that belong exclusively to their respective sets. For example, $(A - B)$ includes elements in set $A$ but not in set $B$, while $(B - A)$ consists of elements in $B$ but not in $A$. Intuitively, since these sets are constructed by removing common elements, trying to intersect them should yield no common elements, represented mathematically as an empty set $\emptyset$.

Understanding this proof nurtures a deeper comprehension of how set operations interact, and emphasizes different ways of applying logical reasoning to problem-solving in set theory. Additionally, it encourages students to think critically about how the properties of set operations such as intersection and difference can lead to elegant solutions to seemingly complex problems, helping solidify foundational knowledge in discrete mathematics.