Final Answer:
The expression (A' ∪ B)' ∩ (B ∪ A)' simplifies to the empty set ∅.
Step-by-step explanation:
The expression (A' ∪ B)' ∩ (B ∪ A)' involves set operations of complement, union, and intersection. Let's break this down step by step.
First, consider A' ∪ B. This represents the complement of set A combined with set B. Taking the complement of A means including all elements not in A, and then the union with set B combines these elements.
Next, the complement of this union [(A' ∪ B)]' means considering all elements not in [(A' ∪ B)]. This operation essentially returns elements that belong to neither the complement of A nor B.
Now, the expression B ∪ A denotes the union of sets B and A.
Finally, the intersection of [(A' ∪ B)]' and (B ∪ A)' means finding the common elements between the complements of [(A' ∪ B)] and (B ∪ A). However, upon simplification and closer inspection, we realize that the complements [(A' ∪ B)]' and (B ∪ A)' contain mutually exclusive sets, resulting in no common elements when intersected.
In essence, since the complements involve disjoint sets, their intersection results in an empty set ∅. This signifies that the final expression (A' ∪ B)' ∩ (B ∪ A)' simplifies to an empty set, meaning there are no elements that satisfy the conditions of this expression, confirming its emptiness.