To solve the given expression $(A - B) \cup (A \cup B)' = B'$, let’s analyze each component step-by-step.

Step-by-Step Solution

1. Interpret $A - B$: $A - B = A \cap B'$ This represents the elements in $A$ but not in $B$.

2. Interpret $(A \cup B)'$: $(A \cup B)' = A' \cap B'$ This is the complement of the union of $A$ and $B$, meaning it contains elements that are in neither $A$ nor $B$.

3. Interpret $(A - B) \cup (A \cup B)'$: Substitute from steps 1 and 2: $(A - B) \cup (A \cup B)' = (A \cap B') \cup (A' \cap B')$ Using the distributive property of union and intersection: $= (A \cup A') \cap B' = U \cap B' = B'$ where $U$ represents the universal set, as $A \cup A' = U$.

Conclusion

$(A - B) \cup (A \cup B)' = B'$ Thus, the equation holds true as both sides are equal.

Would you like further details on any step, or have any other questions?

Here are some related questions to deepen understanding:

1. What is the difference between $A - B$ and $B - A$ in set operations?
2. How can we interpret $(A \cup B)'$ in a Venn diagram?
3. What properties of set operations were used in the solution above?
4. How would the solution change if we had $A \cap B$ instead of $A - B$?
5. How does De Morgan's Law apply to set complements and unions?

Tip: When working with set expressions, always remember to apply De Morgan's Laws when dealing with complements of unions and intersections.